Complexity Results for Preference Aggregation over (m)CP-nets: Pareto and Majority Voting

Combinatorial preference aggregation has many applications in AI. Given the exponential nature of these preferences, compact representations are needed and ($m$)CP-nets are among the most studied ones. Sequential and global voting are two ways to aggregate preferences over CP-nets. In the former, preferences are aggregated feature-by-feature. Hence, when preferences have specific feature dependencies, sequential voting may exhibit voting paradoxes, i.e., it might select sub-optimal outcomes. To avoid paradoxes in sequential voting, one has often assumed the $\mathcal{O}$-legality restriction, which imposes a shared topological order among all the CP-nets. On the contrary, in global voting, CP-nets are considered as a whole during preference aggregation. For this reason, global voting is immune from paradoxes, and there is no need to impose restrictions over the CP-nets' topological structure. Sequential voting over $\mathcal{O}$-legal CP-nets has extensively been investigated. On the other hand, global voting over non-$\mathcal{O}$-legal CP-nets has not carefully been analyzed, despite it was stated in the literature that a theoretical comparison between global and sequential voting was promising and a precise complexity analysis for global voting has been asked for multiple times. In quite few works, very partial results on the complexity of global voting over CP-nets have been given. We start to fill this gap by carrying out a thorough complexity analysis of Pareto and majority global voting over not necessarily $\mathcal{O}$-legal acyclic binary polynomially connected (m)CP-nets. We settle these problems in the polynomial hierarchy, and some of them in PTIME or LOGSPACE, whereas EXPTIME was the previously known upper bound for most of them. We show various tight lower bounds and matching upper bounds for problems that up to date did not have any explicit non-obvious lower bound.

In computer science, the study of preference aggregation has often been based on the solid ground of social choice theory, which is the branch of economics analyzing methods for collective decision making [2,3]. Having a well-founded theory and practice on how to properly and efficiently manage and aggregate preferences of real software agents, and hence support the growth and use of these technologies, has been one of the main drivers for investigating social choice theory from a computational perspective. In social choice theory, the actual ways of representing agent preferences are rarely taken into consideration, also because the sets of candidates usually considered are relatively small in size. For this reason, most of the insights obtained in the computational social choice literature about the computational properties of preference aggregation functions (or voting procedures) have assumed that agent preferences over the set of candidates are extensively listed (see [17] and references therein). Although this is perfectly reasonable when we reason about, e.g., (political) elections among a not too numerous set of human candidates, this is not feasible when the voting domain (i.e., the set of candidates) Figure 2: Dinner preferences of Alice, Bob, and Chuck (in this order) modeled via CP-nets (above) and their extended preference graphs (below). mCP-nets [64], which is a set, or profile, of CP-nets, one for each agent. The preference semantics of mCP-nets is defined via voting schemes: through its own individual CP-net, every agent votes whether an outcome is preferred to another. Various voting schemes were proposed for mCP-nets [52,64], and different voting schemes give rise to different dominance semantics for mCP-nets. In this paper, we consider Pareto and majority voting as they were defined in [64]. In the voting schemes proposed for mCP-nets, the voting protocol adopted, i.e., the actual way in which votes are collected [19], is global voting [46,48]. In this protocol, the results of the voting procedure are computed by having as input the CP-nets as a whole (see Section 6.2 for related works on different voting protocols over CP-nets).

Majority
Majority-Dominance NP-complete Is-Majority-Optimal co-NP-complete Exists-Majority-Optimal Σ P 2 -complete Is-Majority-Optimum Π P 2 -complete Exists-Majority-Optimum Π P 2 -hard, in D P 2 Figure 3: Summary of the complexity results obtained in this paper for global voting over CP-nets. Membership results above P are valid for any representation scheme whose dominance test is feasible in NP. * A different proof is available in [64].

Preliminaries
In this section, we give some preliminaries, briefly recalling from the literature preference relations and aggregation, conditional preference nets (CP-nets), CP-nets for groups of m agents (mCP-nets), and the complexity classes that we will encounter in our complexity results. We also define a formal framework for preference representation schemes, because our membership results will be given for generic representations whose dominance test is feasible in NP.

Preference relations and aggregation
Before dwelling upon the details of CP-nets, which is the specific preference representation analyzed in this paper, we now give an introductory overview of the general concepts of preferences and their aggregation.
In this paper, a preference relation R over a set of outcomes O is a strict order over O, i.e., R is a binary relation over O that is irreflexive (i.e., α, α / ∈ R), asymmetric (i.e., if α, β ∈ R, then β, α / ∈ R), and transitive (i.e., if α, β ∈ R and β, γ ∈ R, then α, γ ∈ R). A preference ranking R is a preference relation that is total (i.e., either α, β ∈ R or β, α ∈ R for any two different outcomes α and β). Usually, given two outcomes α and β, their preference relationship stated in R is denoted by α R β, instead of α, β ∈ R, which means that, in R, α is strictly preferred to β, or α dominates β. On the other hand, α R β means that α, β / ∈ R, and α R means that α, β / ∈ R and β, α / ∈ R, i.e., α and β are incomparable in R. Observe that in a preference ranking, it cannot be the case that two outcomes are incomparable. Given a preference relation R, an outcome α is optimal in R if there is no outcome β such that β R α. We say that α is optimum in R, if for all outcomes β such that β = α, it holds that α R β. Clearly, if there is an optimum outcome in R, then it is unique. For notational convenience, if the preference relation R is clear from the context, we do not explicitly mention R as a subscript in the notations above. In the following, if not stated otherwise, when we speak of preferences structures, we mean preference relations.
In preference aggregation, we deal with preferences of multiple agents. A preference profile P = R 1 , . . . , R m is a set of m preference relations. We assume that all the preferences R i of P are defined over the same set of outcomes, i.e., the agents express their preferences over the same set of candidates. In this paper, we focus on voting procedures based on comparisons of pairs of outcomes (see, e.g., [4] for a classification of different kinds of preference aggregation procedures). For this reason, we need to define the following sets of agents. For a profile P = R 1 , . . . , R m , we denote by S P (α, β) = {i | α Ri β}, S ≺ P (α, β) = {i | α ≺ Ri β}, and S P (α, β) = {i | α Ri β}, the sets of agents preferring α to β, preferring β to α, and for which α and β are incomparable, respectively.
The voting schemes considered in this paper are Pareto and majority. The definition of their dominance semantics over preference profiles, reported below, is a generalization of the respective definition over mCP-nets given in [64].
For a preference profile P and a voting scheme s, if outcome β does not s dominate outcome α, we denote this by β s P α. An outcome α is s optimal in P, if for all β = α, it holds that β s P α, while α is s optimum in P, if for all β = α, it holds that α s P β. Note that optimum outcomes, if they exist, are unique.

CP-nets
We now focus on CP-nets, which is the preference representation that we will more closely investigate in this work. As mentioned in the introduction, the set of outcomes of a preference relation is often defined as the Cartesian product of finite value domains for each of a set of features. Conditional preference nets (CP-nets) [8] are a formalism to encode conditional ceteris paribus preferences over such combinatorial domains. The distinctive element of CP-nets is that a directed graph, whose vertices represent the features of a combinatorial domain, is used to intuitively model the conditional part of conditional ceteris paribus preference statements. Below, we recall the syntax, semantics, and some properties of CP-nets; see Section 6.1 for more on conditional ceteris paribus preferences and preference representations in general.

Syntax of CP-nets
is a directed graph whose vertices F N represent the features of a combinatorial domain, and Dom N and (CPT F N ) F ∈F N are a function and a family of functions, respectively. The function Dom N associates a (value) domain Dom N (F ) with every feature F , while the functions CPT F N are the CP tables for every feature F , which are defined below. The value domain of a feature F is the set of all values that F may assume in the possible outcomes. In this paper, we assume features to be binary, i.e., the domain of each feature F contains exactly two values, usually denoted f and f , and called the overlined and the non-overlined value (of F ), respectively. For a set of features S, Dom N (S) = × F ∈S Dom N (F ) denotes the Cartesian product of the domains of the features in S. Thus, an outcome is an element of O N = Dom N (F N ). Given a feature F and an outcome α, we denote by α[F ] the value of F in α, while, given a set of features F, α [F] is the projection of α over F. For two outcomes α and β, and a set of features F, we denote by α [F] when this is not the case, i.e., when there is at least one feature F ∈ F such that α[F ] = β [F ]. The CP tables encode preferences over feature values. Intuitively, the CP table of a feature F specifies how the values of the parent features of F influence the preferences over the values of F . More formally, for a feature F , we denote by Par N (F ) = {G ∈ F N | G, F ∈ E N } the set of all features in G N from which there is an edge to F . We call Par N (F ) the set of the parents of F (in N ). We denote by Ord N (F ) the set of all the (strict) preference rankings over the elements of Dom N (F ). Each function CPT F N : Dom N (Par N (F )) → Ord N (F ) maps every element of Dom N (Par N (F )) to a (strict) preference ranking over the domain of F . If Par N (F ) = ∅, then CPT F N is a single (strict) preference ranking over Dom N (F ). Note that indifferences between feature values are not admitted in (classical) CP-nets. Each function CPT F N is represented via a two-column table, in which, given a row, the element in the first column is the input value of the function CPT F N , and the element in the second column is the associated (strict) preference ranking over Dom N (F ). Since CPT F N is total, in the table representing the function, there is a row for any combination of values of the parent features, i.e., for a feature F , there are 2 |Dom N (Par N (F ))| rows in the table of F .
In the following, when we define CP tables, we often use a logical notation to identify for which specific values of the parent features, a particular row in the CP table has to be considered. Although this is the notation on which generalized propositional CP-nets [27] are based on, it is used here only for notational convenience. In this paper, we always assume that CP tables are explicitly represented in the input instances. In the CP tables, f f denotes f being preferred to f . We denote by N the size of CP-net N , i.e., the space in terms of bits required to represent the whole net N (which includes features, edges, feature domains, and CP tables).

Semantics of CP-nets
The preference semantics of CP-nets can be defined in several different but equivalent ways [8]. A first definition has a model-theoretic flavour [8, Definitions 2 and 3]. Intuitively, a preference ranking R violates a CP-net N , if there are two outcomes α and β that according to the CP tables of N should be ranked β α, but they are not ranked in such a way in R (i.e., α R β, since R is total is an improving flip. In the following, we often omit the feature F and simply write α → N β; and when we say that we flip a feature, then we often mean that the flipping is improving. The (extended) preference graph of N is the pair G N = V N , E N , where the nodes V N are all the possible outcomes of N , and, given two outcomes α, β ∈ V N , the directed edge from α to β belongs to E N if and only if α → N β.
It can be shown that, for a CP-net N and two outcomes α and β, N |= β α if and only if there is a sequence of improving flips from α to β [8,Theorems 7 and 8]. Therefore, for an agent whose preferences are encoded through a CP-net N , we say that the agent prefers β to α, or that β dominates α (in N ), denoted β N α, if N entails β α, or, equivalently, if there is an improving flipping sequence from α to β. If for two outcomes α and β, neither α N β nor β N α, then α and β are incomparable (in N ), denoted α N β (which is equivalent to the existence of preference rankings R 1 and R 2 that both satisfy N such that α R1 β and β R2 α).
Note here that, since there are no indifferences between features values in (classical) CP-nets, for any two outcomes α and β, either one dominates the other, or they are incomparable. Example 2.1. Consider the CP-net N shown in Figure 4. For the outcomes α = abc and β = abc, it holds that β N α, because α − → N [A] β. For the outcomes α = abc and β = abc, it holds that β N α, because there is no path from α to β in G N . However, α N β, because β − → N [C] α, and hence it is not the case that α N β. Consider now the outcomes α = abc and β = abc. Then, β N α by the improving flipping sequence   abc → abc → abc → abc.
Properties of CP-nets A CP-net is binary, if all its features are binary. The indegree of a CP-net N is the maximum number of edges entering in a node of the graph of N . A CP-net N is singly connected, if, for any two distinct features G and F , there is at most one path from G to F in G N . A class F of CP-nets is polynomially connected, if there exists a polynomial p such that, for any CP-net N ∈ F and for any two features G and F of N , there are at most p( N ) distinct paths from G to F in G N . A CP-net N is acyclic, if G N is acyclic. It is well known that acyclic CP-nets N always have a preference ranking satisfying N , their extended preference graph G N is acyclic, the preferences encoded by N are consistent (i.e., there is no outcome α such that α N α), and there is a unique optimum outcome o N dominating all other outcomes (and, clearly, not dominated by any other), which can be computed in polynomial time [8].
It is known that dominance testing, i.e., deciding, for any two given outcomes α and β, whether β α, is feasible in NP over polynomially connected classes of binary acyclic CP-nets [8]. However, it is an open problem whether dominance testing is feasible in NP over non-polynomially-connected classes of binary acyclic CP-nets. Also, the complexity of dominance testing for non-binary CP-nets is currently still open. Whereas dominance testing for the class of acyclic binary singly connected CP-nets whose indegree is at most six is NP-hard [8]-we improve this result in Section 3, requiring only indegree three. Dominance testing is feasible in polynomial time on acyclic binary CP-nets whose graph is a tree or a polytree [8], and it is PSPACE-complete for cyclic CP-nets [27].
In the rest of this paper, we consider only binary acyclic (and often polynomially connected) classes of CP-nets. When the CP-net N is clear from the context, we often omit the subscript "N " from the notations introduced above.

mCP-nets
In this section, we focus on mCP-nets [64], which are a formalism to reason about conditional ceteris paribus preferences when a group of multiple agents is considered. Intuitively, an mCP-net is a profile of m (individual) CP-nets, one for each agent of the group. The original definition of mCP-nets also allows for partial CP-nets. Here, we consider only mCP-nets consisting of a collection of standard CP-nets. The difference is that we do not allow for non-ranked features in agents' CP-nets, and hence there is no distinction between private, shared, and visible features (see [64] for definitions), i.e., all features are ranked in all the individual CP-nets of an mCP-net.
As underlined in [64], the "m" of an mCP-net stands for multiple agents and also indicates that the preferences of m agents are modeled, so a 3CP-net is an mCP-net with m = 3. Formally, an mCP-net M = N 1 , . . . , N m consists of m CP-nets N 1 , . . . , N m , all of them defined over the same set of features, which, in turn, have the same domains. If M is an mCP-net, we denote by F M the set of all features of M, and by Dom M (F ) the domain of feature F in M. Given this notation, F Ni = F M , for all 1 ≤ i ≤ m, and Dom Ni (F ) = Dom M (F ), for all features F ∈ F M and all 1 ≤ i ≤ m. Although the features of the individual CP-nets are the same, their graphical structures may be different, i.e., the edges between the features in the various individual CP-nets may vary. We underline here that, unlike in other papers in the literature, we do not impose that the individual CP-nets of the agents share a common topological order (i.e., we do not restrict the profiles of CP-nets to be O-legal); see Section 6 for more on O-legality.
An outcome for an mCP-net is an assignment to all the features of the CP-nets, and given an mCP-net M, we denote by O M the set of all the outcomes in M. The preference semantics of mCP-nets is defined through global voting over CP-nets. In particular, via its own individual CP-net, each agent votes whether an outcome dominates another, and hence different ways of collecting votes (i.e., different voting schemes) give rise to different group dominance semantics for an mCP-net. Let M = N 1 , . . . , N m be an mCP-net, and let α and β be two outcomes. With a notation similar to the one defined above, S M (α, β) = {i | α Ni β}, S ≺ M (α, β) = {i | α ≺ Ni β}, and S M (α, β) = {i | α Ni β} are the sets of the agents of M preferring α to β, preferring β to α, and for which α and β are incomparable, respectively.
In [52,64], various voting schemes were proposed and analyzed to define multi-agent dominance semantics for mCP-nets. In this paper, we focus on two of them, namely, Pareto and majority voting, whose dominance semantics definitions are the natural specializations to mCP-nets of Pareto and majority dominance semantics defined above. Consider an mCP-net M = N 1 , . . . , N m , and let α and β be two outcomes. Then: Majority: β majority dominates α, denoted β maj M α, if the majority of the agents of M prefers β to α, i.e., For a voting scheme s, s optimal and s optimum outcomes in mCP-nets are defined in the natural way. An mCP-net is acyclic, binary, and singly connected, if all its CP-nets are acyclic, binary, and singly connected, respectively. A class F of mCP-nets is polynomially connected, if the set of CP-nets constituting the mCP-nets in F is a polynomially connected class of CP-nets. The indegree of an mCP-net is the maximum indegree of its constituent individual CP-nets. Unless stated otherwise, we consider only polynomially connected classes of acyclic binary mCP-nets. When the mCP-net M is clear from the context, we often omit the subscript "M" from the above notations.

Computational complexity
We now give some notions from computational complexity theory, which will be required for the complexity analysis carried out in this paper. First, we briefly recall the complexity classes that we will encounter in this paper (along with some closely related ones), and then we recall the notion of polynomial-time reductions among decision problems, and some decision problems that are hard for some of these complexity classes. We assume that the reader has some elementary background in computational complexity theory, including the notions of Boolean formulas and quantified Boolean formulas, Turing machines, and hardness and completeness of a problem for a complexity class, as can be found, e.g., in [40,60].

Complexity classes
The class P is the set of all decision problems that can be solved by a deterministic Turing machine in polynomial time with respect to the input size, i.e., with respect to the length of the string that encodes the input instance. For a given input string s, its size is usually denoted by s . The class of decision problems that can be solved by nondeterministic Turing machines in polynomial time is denoted by NP. They enjoy a remarkable property: any "yes"-instance s has a certificate for being a "yes"-instance, which has polynomial length and can be checked in deterministic polynomial time (in s ). For example, deciding whether a Boolean formula φ(X) over the Boolean variables X = {x 1 , . . . , x n } is satisfiable, i.e., whether there exists some truth assignment to these variables making φ true, is a well-known problem in NP; in fact, any satisfying truth assignment for φ is clearly a certificate that φ is a "yes"-instance, i.e., that φ is satisfiable.
For a complexity class C, we denote by co-C the complementary class to C, i.e., the class containing the complementary languages of those in C. For example, the problem of deciding whether a Boolean formula φ is not satisfiable is in co-NP. The class P is contained in both NP and co-NP, i.e., P ⊆ NP ∩ co-NP.
By LOGSPACE, we denote the set of decision problems that can be solved by deterministic Turing machines in logarithmic space. For such machines, it is assumed that the input tape is read-only, and that these machine have a read/write tape, called work tape, for intermediate computations. The logarithmic space bound is given on the space available on the work tape. The class LOGSPACE is contained in P.
The class D P , defined originally in [61], is the class of problems that are a "conjunction" of two problems, one from NP and one from co-NP, i.e., D P = {L | L = L ∩ L , L ∈ NP, L ∈ co-NP}. The class co-D P is the class of problems whose complements are in D P , equivalently, it can be defined as the class of problems that are a "disjunction" of two problems, one from NP and one from co-NP, i.e., co-D P = {L | L = L ∪ L , L ∈ NP, L ∈ co-NP}.
The classes Σ P k , Π P k , and ∆ P k , forming the polynomial hierarchy (PH) [71], are defined as follows: Σ P 0 = Π P 0 = ∆ P 0 = P, and, for all k ≥ 1, Σ P k = NP Σ P k−1 , ∆ P k = P Σ P k−1 , and Π P k = co-Σ P k . Here, Σ P k (resp., ∆ P k ) is the set of decision problems solvable by nondeterministic (resp., deterministic) polynomial-time Turing machines with an oracle to recognize, at unit cost, a language in Σ P k−1 . Note that Σ P 1 = NP Σ P 0 = NP P = NP, Π P 1 = co-Σ P 0 = co-NP, and ∆ P 1 = P Σ P 0 = P P = P. Sometimes a bound is imposed on the number of calls that are allowed to be issued to the oracle. For example, Θ P k = P Σ P k−1 [O(log n)] denotes the set of decision problems solvable by a deterministic polynomial-time Turing machine that is allowed to query a Σ P k−1 oracle at most logarithmically many times (in the size of the input). By definition, Θ P k ⊆ ∆ P k . 1 The classes D P and co-D P can be generalized to the classes D P k = {L | L = L ∩ L , L ∈ Σ P k , L ∈ Π P k } and co-D P k = {L | L = L ∪ L , L ∈ Σ P k , L ∈ Π P k }, respectively, for k ≥ 1, that are the conjunction and the disjunction, respectively, of Σ P k and Π P k ; in particular, D P 1 = D P . Note also that D P k ⊆ Θ P k+1 . Given their definitions, for all k ≥ 1, the relationships among the mentioned classes are as follows (see Figure 5 for an illustration): ) (see, e.g., [72,73]). is defined, and s is a "yes"-instance of L 1 if and only if h(s) is a "yes"-instance of L 2 . A decision problem L 1 is polynomially (Karp) reducible to a decision problem L 2 , denoted L 1 ≤ p L 2 , if there is a polynomial-time (Karp) reduction from L 1 to L 2 . In this paper, we consider only Karp reductions.

Reductions and hard problems
To prove hardness for a complexity class, we show reductions from various problems known to be complete for the complexity classes that they belong to. We next define such problems, so that we can later refer to them by name.
Deciding the satisfiability of Boolean formulas, denoted Sat, is the prototypical NP-complete problem, which remains NP-hard even if only 3CNF formulas are considered [26,41], i.e., Boolean formulas in conjunctive normal form with three literals per clause. The complementary problem Unsat of deciding whether a given Boolean formula is not satisfiable is co-NP-complete. It remains co-NP-hard even if only 3CNF formulas are considered, and it is the equivalent to the problem Taut of deciding whether a 3DNF formula is a tautology. A 3DNF formula is a Boolean formula in disjunctive normal form with three literals per term. CNFs and DNFs are actually linked (see [28,29]).

A framework for preference representation schemes
In this paper, most of the membership results that we will show hold for generic preference representation schemes. For this reason, we now introduce the general framework of representation schemes that we will refer to. Inspired by the concept of compact representations in [18,34,36,48], we define preference representation schemes S as suitable encodings for a class of preference relations, denoted C(S ). Formally, a preference representation scheme S defines a computable representation function ξ S (·) and a computable Boolean function Pref S (·, ·, ·) such that, for any relation R ∈ C(S ), ξ S (R) is the encoding of R according to S , and Pref S (ξ S (R), α, β) evaluates to 1, if α, β ∈ R (i.e., if α R β), and to 0, otherwise. By ξ S (R) , we denote the size of the representation of R via S . Let S 1 and S 2 be two preference representation schemes. We say that S 2 is at least as expressive (and succinct) as S 1 , denoted S 1 e S 2 , if there exists a function f in FP (i.e., computable in deterministic polynomial time) that translates a preference relation ξ S1 (R) represented in S 1 into an equivalent preference relation ξ S2 (R) represented in S 2 , i.e., into a preference relation over the same outcomes and with the same preference relationships between them. More precisely, we require that ξ S2 (R) = f (ξ S1 (R)) and Pref S1 (ξ S1 (R), α, β) = Pref S2 (ξ S2 (R), α, β), for each pair of outcomes α and β. Observe that f belonging to FP entails that there exists a constant c f (depending on f ) such that ξ S2 (R) ≤ ξ S1 (R) c f , i.e., the size of ξ S2 (R) is polynomially bounded in the size of ξ S1 (R). 3 A P-and an NP-representation S is a preference representation scheme whose function Pref S is in P and NP, respectively. For example, the polynomially connected classes of acyclic binary CP-nets are NP-representation schemes.
When a compact representation S is clear from the context, we often simply write R instead of ξ S (R), and α R β and α R β instead of Pref S (ξ S (R), α, β) = 1 and Pref S (ξ S (R), α, β) = 0, respectively. While doing so, we are identifying the preference relation with its actual representation.

Complexity of basic tasks on CP-nets
To precisely characterize the complexity of Pareto and majority voting tasks in Sections 4 and 5, we need to understand how complex is deciding, given a CP-net N and two outcomes α and β, whether α dominates β or whether α and β are incomparable. Here, we prove that the former problem is NP-complete, while the latter is co-NP-complete. To achieve this, after giving some preliminary definitions on how to encode Boolean formulas into CP-nets, we show that deciding the satisfiability of Boolean formula can be reduced to the problem of deciding dominance between outcomes in CP-nets. This allows us to prove the NP-hardness of the dominance test in CP-nets, and the co-NP-hardness of deciding incomparability is shown as a byproduct of this property.

Preliminaries
We first introduce a notation mapping Boolean assignments to outcomes of CP-nets; this notation will frequently be used later in the paper. In particular, to prove the hardness of voting tasks on mCP-nets, we often provide reductions from problems regarding the satisfiability (or validity) of (quantified) Boolean formulas. For this reason, mCP-nets will often have sets of features associated with sets of Boolean variables. For example, for a Boolean formula φ(X) over the set of Boolean variables X = {x 1 , . . . , x n }, we often define an mCP-net as a subset of its set of features. Then, for a (partial or complete) assignment σ X over X, an outcome α σ X of M(φ) encoding σ X over the features set V is such that, for the The values of the features F / ∈ V in α σ X will be specified in each particular case.
We next define formula nets, which will be used in hardness proofs, and which are intuitively CP-nets aiming at having a particular preference relationship between two outcomes depending on the satisfiability of associated Boolean formulas in CNF. This will allow us to show that deciding dominance in CP-nets is NP-hard. 4 . Not all the CP tables are reported in the figure.
Formally, let φ(X) be a Boolean formula in 3CNF defined over the set of Boolean variables X = {x 1 , . . . , x n }, and whose set of clauses is C = {c 1 , . . . , c m }. We often omit the variable set from the notation of the Boolean formula, i.e., we write φ instead of φ(X), if this does not cause ambiguity. We denote by j,k the k-th literal of the j-th clause. From φ(X), we build the CP-net F (φ) in the following way (see Figure 6 for an example).
The features of F (φ) are: • for each variable x i ∈ X, there are features V T i and V F i (called variable features), and we denote by V φ the set of variable features; • for each clause c j ∈ C, there is a feature D j (called clause feature), and we denote by D φ the set of clause features; and • for each literal j,k , there is a feature P j,k (called literal feature), and we denote by P φ the set of literal features.
All features are binary, with the usual notation for their values. When the formula φ is clear from the context, we often omit the subscript "φ" from the notation of the sets of features illustrated above. The edges of F (φ) are: for each literal j,k = x i or j,k = ¬x i , there are edges V T i , P j,k , V F i , P j,k , and P j,k , D j . The CP tables of F (φ) are: • for each clause c j ∈ C, feature D j has the CP table Note that F (φ) is binary, acyclic, singly connected, its indegree is three, and the CP-net can be built in polynomial time in the size of φ.
The following lemma and corollary show an important property of formula nets, which is that φ is satisfiable if and only if a particular outcome dominates others in F (φ). Lemma 3.1. Let φ(X) be a Boolean formula in 3CNF defined over a set X of Boolean variables, and let σ X be an assignment on X. Let α σ X be the outcome of F (φ) encoding σ X on the feature set V, and assigning non-overlined values to all other features, and let β be the outcome assigning overlined values to all and only variable and clause features. Then: Proof (sketch). The idea at the base of this proof is that the CP tables in F (φ) are designed so that the features enact the role of variables, literals, and clauses of a CNF Boolean formula. Details of the proof are at page 31.
Proof. Observe that, when the empty assignment σ X is considered, outcomes α σ X and β σ X of the statement of Lemma 3.1 coincide with outcomes α and β of the statement of this corollary, respectively. Moreover, any assignment over X is an extension of the empty one, and hence Lemma 3.1 applies.

Complexity of dominance, incomparability, and optimality on CP-nets
As mentioned above, via formula nets, it is possible to show the NP-hardness and the co-NP-hardness of dominance and incomparability on CP-nets, respectively. We start by showing the NP-hardness of dominance on CP-nets. More formally, consider the following problem on CP-nets.

Problem: Dominance
Instance: A CP-net N , and two outcomes α, β ∈ O N . Question: Is β N α?
Dominance is known to be feasible in NP for some classes of instances, and in particular it is in NP for polynomially connected classes of acyclic binary CP-nets [8]. For these classes of CP-nets, it was shown that Dominance is NP-hard, and hardness holds even if the considered CP-nets are singly connected and the indegree of each feature in the net is at most six [8]. Moreover, Dominance is feasible in polynomial time on acyclic binary CP-nets whose graph is a tree or a polytree [8]. However, the exact complexity of Dominance for general (non polynomially connected classes of) acyclic binary CP-nets is still an open problem [8], and in particular it is unknown whether it belongs to NP or not.
First, we give an improved result on the hardness of dominance testing in CP-nets. In particular, we show that the NP-hardness holds even if the indegree of the CP-net is three, while the minimum indegree previously required to show the hardness was six [8]. [8]). Let N be a CP-net, and let α, β ∈ O N be two outcomes. Deciding whether β N α is NP-hard. Hardness holds even if N is acyclic, binary, singly connected, and its indegree is three.

Theorem 3.3 (improved over
Proof. To show that Dominance is NP-hard, we prove that Sat ≤ p Dominance. Let φ be a Boolean formula in 3CNF. Consider the CP-net F (φ) (defined in Section 3.1) and outcomes α and β such that in α the values of all features are non-overlined, and in β the values of all and only variable and clause features are overlined. By We now focus on the problem of testing incomparability between outcomes in CP-nets and show its co-NP-completeness. We show hardness via the properties of formula nets. More formally, consider the following problem.
The following theorem shows that deciding whether two outcomes are incomparable in a preference relation represented via an NP-representation scheme is in co-NP. Theorem 3.4. Let R be a preference relation represented via an NP-representation scheme, and let α and β be two outcomes. Deciding whether α R β is feasible in co-NP.
Proof. We show that disproving α β is feasible in NP. In fact, if α β, then either α β or β α. In these cases, since R is represented via an NP-representation, there is a polynomial certificate either witnessing α β or witnessing β α. To conclude, observe that such a certificate can be checked in polynomial time.
We now focus on CP-nets and show that deciding incomparability is co-NP-hard. Theorem 3.5. Let N be a CP-net, and let α and β be two outcomes. Deciding whether α N β is co-NP-hard. Hardness holds even if N is acyclic, binary, singly connected, and its indegree is three.
Proof. To show that Incomparability is co-NP-hard, we prove that Unsat ≤ p Incomparability. Let φ be a Boolean formula in 3CNF. Consider the CP-net F (φ) (defined in Section 3.1) and outcomes α and β such that in α the values of all features are non-overlined, and in β the values of all and only variable and clause features are overlined. By Corollary 3.2, φ is unsatisfiable if and only if α F(φ) β.
By combining the previous two results, we obtain that testing incomparability over polynomially connected classes of acyclic CP-nets is co-NP-complete. Corollary 3.6. Let C be a polynomially connected class of acyclic CP-nets. Let N ∈ C be a CP-net, and let α and β be two outcomes. Deciding whether α N β is co-NP-complete.
We emphasize here that checking the incomparability between two outcomes is different from deciding the "ordering query" defined in [8]. For a CP-net N and two outcomes α and β, an ordering query is deciding whether there is at least a preference raking R satisfying N such that α R β. As noticed in [8], this is tantamount to decide whether N |= β α. Since N |= β α if and only if β N α, deciding an ordering query is actually co-NP-hard, because dominance testing is NP-hard. Therefore, it comes as no surprise that the polynomial algorithm proposed in [8] to decide ordering queries is actually "partially complete" (as said in [8]). In fact, given the co-NP-hardness of the ordering query problem, there is no sound and complete deterministic polynomial-time algorithm for this problem (unless a major breakthrough in complexity theory occurs, showing that P = NP).
We conclude this section by looking at the complexity of deciding whether an outcome is optimal in a CP-net. This result is needed in Section 4 to characterize the complexity of one of the voting tasks of Pareto voting. Here, we show that this problem can be decided in LOGSPACE. More formally, consider the following problem.
Problem: Optimality-Testing Instance: A CP-net N , and an outcome α ∈ O N . Question: Is α optimal in N ?
Recall that for acyclic CP-nets, there is an outcome that is optimum [8]. Clearly, this outcome is also the only optimal one in a CP-net. Therefore, checking whether an outcome is optimal is tantamount to checking whether the outcome is optimum. It was shown that, given an acyclic CP-net N , computing the unique optimal outcome o N of N is feasible in deterministic polynomial time, more precisely in linear time, through the "forward sweep" procedure [8]. Hence, as pointed out in [64], a simple procedure to decide whether a given outcome α is optimal in an acyclic CP-net N is to compute o N (in polynomial time) and then to compare α to o N . However, this problem actually belongs to a complexity class that is a below P when acyclic CP-nets are considered. Theorem 3.7. Let N be an acyclic CP-net, and let α ∈ O N be an outcome. Deciding whether α is optimal in N is feasible in LOGSPACE.
Proof. Being N acyclic, if an outcome α is not optimal in N , then there is an improving flipping sequence from α to the optimum outcome, and hence there is at least a feature whose value can be flipped in α to obtain a better outcome. Therefore, to decide whether α is optimal, it suffices to consider in turn all features F and check whether it is possible to perform an improving flip according to the CP table of F . If no feature can be flipped to improve the outcome, then α is optimal. Clearly, this procedure requires only logarithmic space to be carried out. Now that we have analyzed the complexity of dominance, incomparability, and optimality in CP-nets, we can devote our focus to the complexity of Pareto and majority voting on CP-nets in the next two sections.

Complexity of Pareto voting on mCP-nets
In this section, we characterize the complexity of Pareto voting tasks on mCP-nets. In particular, after giving some preliminaries on specific structures of CP-nets that we will use in our reductions, we analyze the complexity of Pareto dominance, which is proven NP-complete. Then, we devote our analysis to the problems related to Pareto optimal outcomes, namely, deciding whether an outcome is Pareto optimal, and deciding whether an mCP-net has a Pareto optimal outcome. We prove the former co-NP-complete, while for the latter, we are able to show that every mCP-net has a Pareto optimal outcome, which implies that the problem is trivial (i.e., feasible in constant time). To conclude, we study the complexity of problems on Pareto optimum outcomes, namely, deciding whether an outcome is Pareto optimum, and deciding whether an mCP-net has a Pareto optimum outcome. Both problems are proven to be tractable, and in particular we show the former to be in feasible in LOGSPACE and the latter to be feasible in P. We recall that the Pareto voting semantics is based on the concept of unanimity, i.e., given an mCP-net M and two outcomes α,

Preliminaries
We now introduce a specific structure of CP-nets that will be used in the forthcoming reductions. In particular, the conjunctive H C (·) and disjunctive H D (·) interconnecting nets are CP-nets whose role is intuitively to link different parts of bigger CP-nets: given a set S of m features, the aim of the CP-nets H C (m) and H D (m) is to propagate the information that all features of S and at least one of the features of S, respectively, have been flipped to their overlined value.
We first introduce the conjunctive interconnecting CP-net H C (m). See Figure 7 for an example of a H C (9) interconnecting net. Such a CP-net is partitioned into layers, and it is a kind of "inverted pyramid". In particular, H C (m) is an acyclic DAG in which each feature of a layer has two or three distinct parents in the previous layer, and at most one child in the next layer. Features belonging to the same layer have no parents in common, and in every layer at most one feature has three parents. The first layer of H C (m) is attached to a set S of m different features of the CP-net that we want to interconnect. In the first layer, the above described connection properties hold relative to the features of S. The layer with a unique feature, which we call apex, is the last layer of the CP-net. All the features of H C (m) are named A i with a proper increasing index i, and their values are, as usual, {a i , a i }. The CP table for a feature A i states that value a i is preferred to a i whenever the value of every parent of A i is overlined. Otherwise, the value a i is preferred to a i . Let α be the outcome in which the values of all the features A i are non-overlined. It is not difficult to see that, whenever all the features in S have overlined values in α, there is an improving flipping sequence starting from α changing the values of all A i (and hence also the value of the apex) to their overlined values.
The disjunctive interconnecting net H D (m) is similar to H C (m). The features and the structure of H D (m) are the same as those in H C (m). The only variations are on the CP tables. Being H D (m) a disjunctive net, given a feature A i of H D (m), value a i is preferred to value a i , whenever at least one of the parents of A i has been flipped to its overlined value. Let α be the outcome in which the values of all the features A i are non-overlined. It is easy to see that, whenever at least a feature in S has an overlined value in α, there is an improving flipping sequence starting from α and reaching an outcome in which the apex has an overlined value.
Note that H C (m) and H D (m) are binary, acyclic, singly connected, their indegree is at most three, and the CP-nets can be built in polynomial time in |S| = m, as the number of their features is polynomial in m (in particular, strictly less than m), and each feature has a bounded number of parents which translates into CP tables of bounded sizes.

Complexity of Pareto dominance on mCP-nets
First, we analyze the problem of deciding Pareto dominance on CP-nets, which is shown NP-complete. More formally, consider the following problem.

Problem: Pareto-Dominance
Instance: An mCP-net M, and two outcomes α, β ∈ O M . Question: Is β p M α? The following theorem shows that deciding Pareto dominance over preference profiles represented via an NP-representation scheme is in NP.
Theorem 4.1. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme, and let α and β be two outcomes. Then, deciding whether β p P α is feasible in NP. Proof. To show that this problem resides in NP, we exhibit a concise certificate for it. Let P = R 1 , . . . , R m . If β p P α, then, for all 1 ≤ i ≤ m, it holds that β Ri α. Since, for every agent i, R i is represented via an NP-representation scheme, there is a concise certificate witnessing that β Ri α. Therefore, in order to decide whether β p P α, it suffices to guess and subsequently check the polynomial witnesses that, for each agent i, it holds that β Ri α. The overall guess requires only polynomial space, and it can be checked in polynomial time.
Observe that on 1CP-nets, Pareto dominance is equivalent to dominance (on simple CP-nets). Therefore, the following result, which follows directly from Theorem 3.3, shows that deciding Pareto dominance in mCP-nets is NP-hard.

Theorem 4.2.
Let M be an mCP-net, and let α and β be two outcomes. Then, deciding whether β p M α is NP-hard. Hardness holds even on classes of singly connected acyclic binary mCP-nets with indegree at most three and at most one agent.
By combining the two previous results, we immediately obtain that Pareto dominance over polynomially connected classes of acyclic mCP-nets is NP-complete.

Complexity of Pareto optimality on mCP-nets
Here, we devote our analysis to problems on Pareto optimal outcomes. In particular, the problems analyzed are deciding whether an outcome is Pareto optimal, and deciding whether an mCP-net has a Pareto optimal outcome. We first focus on deciding Pareto optimality of outcomes in mCP-nets. We show that this problem is co-NP-complete. More formally, consider the following problem.
Problem: Is-Pareto-Optimal Instance: An mCP-net M, and an outcome α ∈ O M . Question: Is α Pareto optimal in M?
The following result shows that, for a preference profile represented via an NP-representation scheme, deciding the Pareto optimality of an outcome is feasible in co-NP. Theorem 4.4. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme, and let α be an outcome. Then, deciding whether α is Pareto optimal in P is feasible in co-NP.
Proof. We show that disproving α being Pareto optimal is feasible in NP. If α is not Pareto optimal in P, then there is an outcome β such that β p P α. Therefore, we can guess such an outcome β along with the witness that β p P α. This guess requires only polynomial space and can be checked in polynomial time (see the proof of Theorem 4.1).
To prove the co-NP-hardness of Is-Pareto-Optimal, we use a reduction from Unsat. Consider the following construction. Let φ(X) be a Boolean formula in 3CNF defined over the set of Boolean variables X = {x 1 , . . . , x n }, and whose set of clauses is in the following way. The CP-nets N ipo 1 and N ipo 2 are built similarly, and we discuss first N ipo 1 . See Figure 8 for a schematic representation of the interconnections between the building blocks of N ipo 1 and N ipo 2 . In N ipo 1 , there are two complete copies of the net F (φ) (defined in Section 3.1), with its features, edges, and CP tables. To distinguish these two copies, we append two different superscript to them and obtain F (φ) a and is similar to N ipo 1 , with the only difference that the conjunctive interconnecting CP-net H C (m) attaches the clause features of F (φ) b to the variable features of F (φ) a . The CP tables in N ipo 2 of the variable features of F (φ) a are therefore modified accordingly.
Observe that M ipo (φ) is acyclic, binary, its indegree is three, and can be computed in polynomial time from φ. Moreover, the class of mCP-nets {M ipo (φ)} φ derived from formulas φ of the specified kind and according to the reduction shown above is polynomially connected. The following result shows that φ(X) is satisfiable if and only if a particular outcome of M ipo (φ) is not Pareto optimal. The above property implies that, in mCP-nets, deciding the Pareto optimality of an outcome is co-NP-hard. Theorem 4.6. Let M be an mCP-net, and let α ∈ O M be an outcome. Deciding whether α is Pareto optimal is co-NP-hard. Hardness holds even on polynomially connected classes of acyclic binary mCP-nets with indegree at most three and at most two agents.
Proof. We prove that Is-Pareto-Optimal is co-NP-hard by showing a reduction from Unsat. Let φ(X) be a 3CNF formula, and consider the 2CP-net M ipo (φ) = N ipo 1 , N ipo 2 . Consider the outcome α ∈ O Mipo(φ) in which the values of all features are non-overlined. By Lemma 4.5, φ(X) is unsatisfiable (and hence a "yes"-instance of Unsat) if and only if α is Pareto optimal in M ipo (φ).
Notice here that the presence of at least two agents in an mCP-net is an essential source of complexity for the problem Is-Pareto-Optimal. In fact, if there were only one agent, then deciding whether an outcome α is Pareto optimal would be tantamount to checking whether α is optimal for that only agent, and we have seen already that this task can be carried out in LOGSPACE (see Theorem 3.7).
By combining the two above results, we immediately conclude that deciding the Pareto optimality of an outcome over polynomially connected classes of acyclic mCP-nets is co-NP-complete. Corollary 4.7. Let C be a polynomially connected class of acyclic mCP-nets. Let M ∈ C be an mCP-net, and let α ∈ O M be an outcome. Deciding whether α is Pareto optimal is co-NP-complete.
We now focus on the problem of deciding the existence of Pareto optimal outcomes in mCP-nets. We show that every mCP-nets has a Pareto optimal outcome, which implies that the problem is trivial (i.e., feasible in constant time). More formally, consider the following problem.
Problem: Exists-Pareto-Optimal Instance: An mCP-net M. Question: Does M have a Pareto optimal outcome?
The following lemma shows that an acyclic CP-net has always a Pareto optimal outcome. Its proof is different from the proof of a similar result that appeared in [64].

Complexity of Pareto optimums on mCP-nets
We now focus on Pareto optimum outcomes. In particular, the problems analyzed are deciding whether an outcome is Pareto optimum, and deciding whether an mCP-net has a Pareto optimum outcome. To study the complexity of these problems, we first prove an intermediate property stating that an mCP-net has a Pareto optimum outcome if and only if all its individual CP-nets have the very same optimum outcome. Proof. We show first that if M = N 1 , . . . , N m has a Pareto optimum outcome α, then all the individual CP-nets have the very same optimum outcome, which is α. By definition, a Pareto optimum outcome is unique and Pareto dominates all other outcomes. This means that, for any outcome β = α, and for all i, α Ni β. We know that in each individual CP-net the only outcome dominating all the others is the individual optimum. Therefore, α equals all the individual optimum outcomes. The other direction of the proof is easy.
Based on the property above, we characterize the complexity of problems on Pareto optimum outcomes. We first focus on deciding whether outcomes are Pareto optimum in mCP-nets. We show that this problem is solvable in LOGSPACE. More formally, consider the following problem.
Problem: Is-Pareto-Optimum Instance: An mCP-net M and an outcome α. Question: Is α Pareto optimum in M?
The following theorem shows that, in acyclic mCP-nets, deciding that an outcome is Pareto optimum is feasible in LOGSPACE. Proof. From Lemma 4.9, we know that α is Pareto optimum in M if and only if α equals all the individual optimum outcomes of the individual CP-nets of M. Therefore, in order to check whether α is actually Pareto optimum it suffices to check, for each individual CP-net in turn, whether α is the individual optimal outcome for that agent. If α is different from even just one of the individual optimum outcomes, then α is not Pareto optimum. Remember that checking the individual optimality of α is feasible in LOGSPACE (see Theorem 3.7), and hence, by reusing of work space, we can check in LOGSPACE whether α equals all the individual optimum outcomes.
To conclude, we study the complexity of deciding whether an mCP-net has a Pareto optimum outcome. We show that this problem is feasible in P. More formally, consider the following problem. The following theorem states that deciding whether an acyclic mCP-net has a Pareto optimum outcome is in P. Proof. From Lemma 4.9, we know that M has a Pareto optimum outcome if and only if all the individual CP-nets have the very same individual optimum outcome. Hence, in order to decide whether M has a Pareto optimum outcome, it suffices to compute the individual optimum outcome of the first agent. We can do this in polynomial time [8]. After this, we compare the just computed individual optimum outcome with all the other individual optimum outcomes. This can be carried out, by reusing of working space, in logarithmic space (see Theorem 4.10), and hence in polynomial time (by the inclusion LOGSPACE ⊆ P). If all the individual optimum outcomes are equal, then we answer yes, otherwise no. Observe that the overall procedure is feasible in P.

Complexity of majority voting on mCP-nets
In this section, we characterize the complexity of majority voting tasks on mCP-nets. We start with a preliminary section by showing that there are mCP-nets without majority optimal and optimum outcomes, which implies that deciding the existence of majority optimal and optimum outcomes is not a trivial problem. In the preliminary section, we also define some CP-nets that will be used in the reductions exhibited in this section. More specifically, we will introduce directs nets, which are CP-nets having a designated outcome as optimal. Moreover, we introduce summarized formula nets, which, similarly to formula nets, are CP-nets encoding Boolean formulas, but they associate the satisfiability of formulas with outcomes having specific values on just two features. Then, we analyze the complexity of deciding majority dominance in mCP-nets, which is shown NP-complete. Subsequently, we devote our analysis to the problems related to majority optimal outcomes, namely, deciding whether an outcome is majority optimal, and deciding whether an mCP-net has a majority optimal outcome. We prove the former co-NP-complete and the latter Σ P 2 -complete. To conclude, we study the complexity of problems on majority optimum outcomes, namely, deciding whether an outcome is majority optimum, and deciding whether an mCP-net has a majority optimum outcome. We prove the former Π P 2 -complete and the latter is shown Π P 2 -hard and belonging to D P 2 . Recall that, given an mCP-net M and two different outcomes α, β ∈ O M , it holds that β maj M α, if the majority of agents prefer β to α, i.e., What we call majority optimal and majority optimum outcomes, in some works (see, e.g., [25,51]) are named weak and (strong) Condorcet winners, respectively. However, in the literature (see, e.g., [4,16,17]), the nomenclature of weak/strong Condorcet winner has also been used with a slightly different meaning. To avoid any confusion, in this paper, we prefer to stick to the concepts of majority optimal and majority optimum outcomes, which we introduced in Section 2.

Preliminaries
We first show that there are mCP-nets that do not have any majority optimal outcome, and hence neither a majority optimum outcome. Thus, deciding whether an mCP-net has majority optimal or optimum outcomes is a non-trivial problem.
Theorem 5.1. There are acyclic binary singly-connected mCP-nets not having majority optimal and majority optimum outcomes.
The preferences encoded in the four nets are: ab N1 ab N1 ab N1 ab; ab N2 ab N2 ab N2 ab; ab N3 ab N3 ab N3 ab; and ab N4 ab N4 ab N4 ab. Observe that: ab is not majority optimal,  because ab maj M NoWin ab; ab is not majority optimal, because ab maj M NoWin ab; ab is not majority optimal, because ab maj M NoWin ab; and ab is not majority optimal, because ab maj M NoWin ab. This implies that M NoWin does not have any majority optimal outcome, and hence also either a majority optimum outcome.
We now define two CP-nets that will be used in the reductions of this section on majority voting. First, we define direct nets, which intuitively are CP-nets that have a specific desired optimum outcome (recall that an acyclic CP-net has a unique optimal outcome, which is also optimum). Let S be a set of binary features defined over the usual values, and let α ∈ Dom(S) be an outcome over S. We finally introduce summarized formula nets, which are similar to formula nets (of Section 3.1), with the advantage that these new nets put in relationship the satisfiability of Boolean formulas with the flip of only two features, instead of with the flip of all variable and clause features. This advantage comes at cost of loosing the single connectedness property of the nets, which, instead, is satisfied in non-summarized formula nets. Formally, let φ(X) be a Boolean formula in 3CNF defined over the set of Boolean variables X = {x 1 , . . . , x n }, and whose set of clauses is C = {c 1 , . . . , c m }. From φ(X), we build the CP-net F s (φ) in the following way (see Figure 11).
The CP-net F s (φ) embeds a formula net F (φ) (defined in Section 3.1) with its features and links. Moreover, in F s (φ), there is an interconnecting net H C (m) (defined in Section 4.1), which is attached to all clause features of F (φ). We denote by A the set of features belonging to the net H C (m) embedded in F s (φ). To conclude with the features of F s (φ), there are two more features: U 1 and U 2 , where U 1 has no parents and is linked to all variable and literal features of F (φ), while U 2 is not the parent of any feature and its unique parent is the apex of the interconnecting net.
The CP tables of F s (φ) are as follows: • feature U 1 has the CP table u 1 u 1 ; • for each variable x i ∈ X, features V T i and V F i have the CP tables • for each literal j,k , if j,k = x i , then feature P j,k has the CP table otherwise, if j,k = ¬x i , then P j,k has the CP table • clause features have the same CP table as in F (φ); • features of the conjunctive interconnecting net H C (m) have the usual CP tables; • feature U 2 , if feature A is the apex of the interconnecting net, has the CP table a u 2 u 2 a u 2 u 2 .
Note that F s (φ) is binary, acyclic, its indegree is three, and the net can be built in polynomial time in the size of φ. Moreover, the class of mCP-nets {M ipo (φ)} φ derived from formulas φ of the specified kind and according to the reduction shown above is polynomially connected. Now we give an equivalent of Lemma 3.1 for F s (φ). In particular, the following lemma and corollary show that φ is satisfiable if and only if a particular outcome dominates others in F s (φ).

Lemma 5.2. Let φ(X) be a Boolean formula in 3CNF defined over a set X of Boolean variables, and let σ X
be an assignment on X. Let α σ X be the outcome of F s (φ) encoding σ X on the feature set V, and assigning non-overlined values to all other features. Let β be an outcome of F s (φ) such that β[U 1 U 2 ] = u 1 u 2 , assigning any value to the features of V, and assigning non-overlined values to all other features. Then: (1) There is an extension of σ X to X satisfying φ(X) if and only if β Fs(φ) α σ X ; (2) There is no extension of σ X to X satisfying φ(X) if and only if β Fs(φ) α σ X .
Proof (sketch). The intuition at the base of the proof of this property is that, by linking U 1 to variable and literal features, these cannot be flipped to their overlined values once U 1 is u 1 . Therefore, distinct literal features, attached to the same features V T i and V F i , cannot be flipped to their overlined values according to contrasting values of V T i and V F i . Details of the proof are at page 32.

• φ(X) is satisfiable if and only if β Fs(φ) α, and • φ(X) is unsatisfiable if and only if β Fs(φ) α.
Proof. Observe that, when the empty assignment σ X is considered, outcome α σ X of the statement of Lemma 5.2 coincides with outcome α of the statement of this corollary. Moreover, any assignment over X is an extension of the empty one, and hence Lemma 5.2 applies.

Complexity of majority dominance on mCP-nets
First, we analyze the problem of deciding majority dominance on CP-nets, which is shown NP-complete. More formally, consider the following problem.
The following result shows that, for preference profiles represented via an NP-representation scheme, deciding majority dominance is feasible in NP.

Theorem 5.4. Let P be a preference profile defined over the same combinatorial domain and represented via
an NP-representation scheme, and let α and β be two outcomes. Then, deciding whether β maj P α is feasible in NP.
Proof. Let P = R 1 , . . . , R m . Observe first that, since |S P (β, α)| + |S ≺ P (β, α)| + |S P (β, α)| = m, it holds that |S P (β, α)| > |S ≺ P (β, α)| + |S P (β, α)| if and only if |S P (β, α)| > m 2 . If β maj P α, then, for more than half of the agents i, β Ri α. For such agents, since the preferences are represented via an NP-representation scheme, there is a polynomial witness that they prefer β to α. Therefore, to show that β maj P α, it suffices to guess a set S of players preferring β to α, along with the polynomial witness of their preference, and then check that |S| > m 2 and that the witnesses are valid. The overall guess requires only polynomial space, and it can be checked in polynomial time.
Observe that, on 1CP-nets, majority dominance is equivalent to dominance on (simple) CP-nets. Therefore, the following result, which follows directly from Theorem 3.3, states that on mCP-nets deciding majority dominance is NP-hard.
Theorem 5.5. Let M be an mCP-net, and let α, β ∈ O M be two outcomes. Then, deciding whether β maj M α is NP-hard. Hardness holds even on classes of singly connected acyclic binary mCP-nets with indegree at most three and at most one agent.
By combining the two above results, we immediately obtain that deciding majority dominance over polynomially connected classes of acyclic mCP-nets is NP-complete. Corollary 5.6. Let C be a polynomially connected class of acyclic mCP-nets. Let M ∈ C be an mCP-net, and let α, β ∈ O M be two outcomes. Then, deciding whether β maj M α is NP-complete.

Complexity of majority optimality on mCP-nets
Here, we analyze the problems on majority optimal outcomes. In particular, the problems considered are deciding whether an outcome is majority optimal, and deciding whether an mCP-net has a majority optimal outcome. We first focus on deciding majority optimality of outcomes in mCP-nets. We show that this problem is co-NP-complete. More formally, consider the following problem.

Problem: Is-Majority-Optimal
Instance: An mCP-net M, and an outcome α ∈ O M . Question: Is α majority optimal in M?
The following theorem shows that, on preference profiles represented via an NP-representation scheme, deciding whether an outcome is majority optimal is feasible in co-NP. Theorem 5.7. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme, and let α be an outcome. Then, deciding whether α is majority optimal in P is feasible in co-NP.
Proof. We show that disproving α being majority optimal is feasible in NP. If α is not majority optimal in P, then there is an outcome β such that β maj P α. Therefore, we can guess such an outcome β along with the witness that β maj P α (i.e., the set of agents preferring β to α). This guess requires only polynomial space, and can be checked in polynomial time (see the proof of Theorem 5.4).
Observe that, on 2CP-nets, majority dominance and Pareto dominance are equivalent. Therefore, the following result, which follows directly from Theorem 4.6, shows that on mCP-nets deciding majority optimality is co-NP-hard.
Theorem 5.8. Let M be an mCP-net, and let α ∈ O M be an outcome. Then, deciding whether α is majority optimal is co-NP-hard. Hardness holds even on polynomially connected classes of acyclic binary mCP-nets with indegree at most three and at most two agents.
By combining the two previous results, it follows immediately that deciding the majority optimality of an outcome over polynomially connected classes of acyclic mCP-nets is co-NP-complete. Corollary 5.9. Let C be a polynomially connected class of acyclic mCP-nets. Let M ∈ C be an mCP-net, and let α ∈ O M be an outcome. Then, deciding whether α is majority optimal is co-NP-complete.
We now focus on the problem of deciding the existence of majority optimal outcomes in mCP-nets. We show that this problem is Σ P 2 -complete. More formally, consider the following problem. The following theorem shows that deciding whether a preference profile represented via an NP-representation scheme has a majority optimal outcome is feasible in Σ P 2 .
Theorem 5.10. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme. Then, deciding whether P has a majority optimal outcome is feasible in Σ P 2 .
Proof. To show that P has a majority optimal outcome, it suffices to guess an outcome α and then check that α is actually majority optimal. Observe that guessing α requires an NP machine, and the final check is feasible in co-NP (see Theorem 5.7), which can be carried out by an oracle. Therefore, the overall procedure is feasible in Σ P 2 .
The features of M eml (Φ) are: • all the features of a net F s (φ) (defined in Section 5.1) in which, in this case, we distinguish two variable recall that P and D are the sets of literal and clause features, respectively, and A is the set of features of the conjunctive interconnecting net embedded in F s (φ)); for further reference, we call A the apex of the interconnecting net; • N eml 2 is similar to net N eml 1 , with the only differences that features U 1 and U 2 are exchanged, and the CP tables of N eml 2 are adjusted to reflect this change.
is as follows (see Figure 12 for a schematic representation of the links). Links of N eml 3 are the following: ; a disjunctive interconnecting H C (|V | + |W| + |P| + |D| + |A|) over feature set B which is connected to the features in V ∪ W ∪ P ∪ D ∪ A; CP tables of N eml 3 are the following: features in B of the interconnecting net have the usual CP tables; -U 2 has the CP table • N eml 4 is equal to net N eml 3 ; • N eml 5 is composed by a link from U 1 to U 2 , and a direct net D(γ), where γ is defined over all features but U 1 and U 2 , and assigns non-overlined values to all of them. The other CP tables of N eml 5 are: feature U 1 has the CP table u 1 u 1 ; feature U 2 has the CP table • N eml 6 is characterized by having a link from feature U 2 to any other feature. The CP tables of N eml 6 are: feature U 2 has the CP table u 2 u 2 ; features F different from U 2 have the CP table Observe that M eml (Φ) is acyclic, binary, its indegree is three, and can be computed in polynomial time from φ. Moreover, the class of mCP-nets {M eml (Φ)} Φ derived from formulas Φ of the specified kind and according to the reduction shown above is polynomially connected. It is possible to show that Φ = (∃X)(∀Y )¬φ(X, Y ) is valid if and only if M eml (Φ) has a majority optimal outcome. Proof (sketch). The intuition at the base of the proof is to put in relationship truth assignments over the variable set X with outcomes of M eml (Φ). In particular, given an assignment σ X over X, the associated outcome is β σ X , where σ X is encoded over the feature set V in the usual way, and all other features have non-overlined values. We show first that outcomes not in the form of a β σ X are not majority optimal. Then, we show that if there is an assignment σ X such that (∀Y )¬φ(X/σ X , Y ) is valid, then β σ X is majority optimal. On the other hand, if there is no assignment σ X for which (∀Y )¬φ(X/σ X , Y ) is valid, then none of the outcomes β σ X is majority optimal. Details of the proof are given at page 34.
We now prove that deciding whether an mCP-net has a majority optimal outcome is Σ P 2 -hard.

Theorem 5.12. Let M be an mCP-net. Then, deciding whether there is a majority optimal outcome in M is Σ P 2 -hard. Hardness holds even on polynomially connected classes of acyclic binary mCP-nets with indegree at most three and at most six agents.
Proof. We prove the hardness of Exists-Majority-Optimal by showing a reduction from QBF CNF 2,∀,¬ . Let Φ = (∃X)(∀Y )¬φ(X, Y ) be an instance of QBF CNF 2,∀,¬ , and consider the 6CP-net M eml (Φ). By Lemma 5.11, Φ is a "yes"-instance of QBF CNF 2,∀,¬ if and only if there is majority optimal outcome in M eml (Φ).
By combining the two above results, we immediately conclude that deciding the existence of majority optimal outcomes over polynomially connected classes of acyclic mCP-nets is Σ P 2 -complete.
Corollary 5.13. Let C be a polynomially connected class of acyclic mCP-nets. Let M ∈ C be an mCP-net. Then, deciding whether there is a majority optimal outcome in M is Σ P 2 -complete.

Complexity of majority optimums on mCP-nets
We now focus on majority optimum outcomes. In particular, the problems analyzed are deciding whether an outcome is majority optimum, and deciding whether an mCP-net has a majority optimum outcome. We first consider the problem of deciding whether an outcome is majority optimum in an mCP-net. We prove that this problem is Π P 2 -complete. More formally, consider the following problem. Problem: Is-Majority-Optimum Instance: An mCP-net M, and an outcome α ∈ O M . Question: Is α majority optimum in M?
The following result shows that, on preference profiles represented via an NP-representations scheme, deciding whether an outcome is majority optimum is feasible in Π P 2 .
Theorem 5.14. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme, and let α be an outcome. Deciding whether α is majority optimum in P is feasible in Π P 2 .
Proof. We prove the statement by showing that deciding whether α is not majority optimum in P is feasible in Σ P 2 . If α is majority optimum, then there is an outcome β such that α maj P β. Therefore, in order to prove that α is not majority optimum, it suffices to guess β, and then check that α maj P β. Observe that guessing β requires an NP machine, and then checking α maj P β is feasible in co-NP (see Theorem 5.4), which can be carried out by an oracle. Therefore, the overall procedure is feasible in Σ P 2 .
To prove the Π P 2 -hardness of Is-Majority-Optimum, we use a reduction from QBF CNF 2,∀,¬ . Consider the following construction. Let Φ = (∃X)(∀Y )¬φ(X, Y ) be a quantified formula where φ(X, Y ) is a 3CNF Boolean formula defined over two disjoint sets X = {x 1 , . . . , x n X } and Y = {y 1 , . . . , y n Y } of Boolean variables, and whose set of clauses is C = {c 1 , . . . , c m }. • all the features of a net F s (φ) (defined in Section 5.1) in which, in this case, we distinguish two variable is as follows (see Figure 12 for a schematic representation of the links of N eml 3 , which is equivalent to N imm 3 ). Links of N imm 3 are the following: ; a disjunctive interconnecting H C (|V | + |W| + |P| + |D| + |A|) over feature set B which is connected to the features in V ∪ W ∪ P ∪ D ∪ A; CP tables of N imm 3 are the following: features in B of the interconnecting net have the usual CP tables; -U 2 has the CP table Observe that M imm (Φ) is acyclic, binary, its indegree is three, and can be computed in polynomial time from φ. Moreover, the class of mCP-nets {M imm (Φ)} Φ derived from formulas Φ of the specified kind and according to the reduction shown above is polynomially connected. The following result shows that Φ = (∃X)(∀Y )¬φ(X, Y ) is valid if and only if a particular outcome of M imm (Φ) is majority optimum. Proof (sketch). The intuition at the base of the proof is to put in relationship truth assignments over the variable set X with outcomes of M imm (Φ). In particular, given an assignment σ X over X, the associated outcome is β σ X , where σ X is encoded over the feature set V in the usual way, and all other features have non-overlined values. We show first that α majority dominates any other outcome β that is not in the form of a β σ X outcome, and hence none of them is majority optimum. Moreover, for all such outcomes β σ X , β σ X does not majority dominates α, and hence, again, none of them is majority optimum. So, only α is candidate to be majority optimum. Then, we show that if there is an assignment σ X such that (∀Y )¬φ(X/σ X , Y ) is valid, then α maj Mimm(Φ) β σ X , and hence α is not majority optimum, which implies that M imm (Φ) does not have any majority optimum outcome. On the other hand, if there is no assignment σ X for which (∀Y )¬φ(X/σ X , Y ) is valid, then, for all the outcomes β σ X , α maj Mimm(Φ) β σ X , which implies that α is majority optimum. Details of the proof are given at page 38.
The next result shows that deciding whether an outcome is majority optimum in an mCP-net is Π P 2 -hard. Theorem 5.16. Let M be an mCP-net, and let α ∈ O M be an outcome. Then, deciding whether α is majority optimum in M is Π P 2 -hard. Hardness holds even on polynomially connected classes of acyclic binary mCP-nets with indegree at most three and at most three agents.
Proof. We prove the statement by showing that deciding whether α is not majority optimum is Σ P 2 -hard, and we do this by exhibiting a reduction from QBF CNF 2,∀,¬ to the complement problem to Is-Majority-Optimum. Let Φ = (∃X)(∀Y )¬φ(X, Y ) be an instance of QBF CNF 2,∀,¬ , and consider the 3CP-net M imm (Φ), and the outcome α in which only the values of features U 1 and U 2 are overlined. By Lemma 5.15, Φ is a "yes"-instance of QBF CNF 2,∀,¬ if and only if α is not majority optimum in M imm (Φ).
Note that, with respect to the number of agents, the above result is optimal. Indeed, since majority dominance and Pareto dominance are equivalent on mCP-nets with m ≤ 2, it is not possible to show the Π P 2 -hardness of Is-Majority-Optimum on mCP-nets with m < 3.
By combining the two previous results, we immediately conclude that deciding whether an outcome is majority optimum over polynomially connected classes of acyclic mCP-nets is Π P 2 -complete.
Corollary 5.17. Let C be a polynomially connected class of acyclic mCP-nets. Let M ∈ C be an mCP-net, and let α ∈ O M be an outcome. Then, deciding whether α is majority optimum in M is Π P 2 -complete.
To conclude, we study the complexity of deciding whether an mCP-net has a majority optimum outcome. We show that the problem is Π P 2 -hard and belongs to D P 2 . More formally, consider the following problem. Problem: Exists-Majority-Optimum Instance: An mCP-net M. Question: Does M have a majority optimum outcome?
We prove that deciding whether a preference profile represented via an NP-representation scheme has a majority optimum outcome is feasible in D P 2 . The following lemma is an intermediate result to show the membership in D P 2 .
Lemma 5.18. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme. Then, deciding whether P does not have majority optimal outcomes that are not also majority optimum is feasible in Π P 2 .
Proof. To prove the statement of the lemma, we show that the complement task to the one in the statement is feasible in Σ P 2 . First, we guess two different outcomes α and β, which is feasible in NP. Then, through a co-NP oracle call we check whether α is actually majority optimal (see Theorem 5.7), then through another oracle call in co-NP we check that α maj P β (see Theorem 5.4). If the answer to the latter oracle call is "yes", then it means that α is a majority optimal outcome that is not also majority optimum (because α does not majority dominate β). To conclude, observe that the overall procedure is feasible in Σ P 2 .
Theorem 5.19. Let P be a preference profile defined over the same combinatorial domain and represented via an NP-representation scheme. Then, deciding whether P has a majority optimum outcome is feasible in D P 2 .
Proof. Let O be a combinatorial domain. Let A be the set of all the preference profiles over O having at least a majority optimal outcome. Let B be the set of all the preference profiles over O not having majority optimal outcomes that are not also majority optimum (observe that in B there might be profiles with no majority optimal outcome at all, and profiles having a majority optimal outcome that is also majority optimum). Clearly, the intersection A ∩ B is the set of all the preference profiles over O having a majority optimum outcome. Observe that, if we focus only on NP-representations schemes, then set A can be decided in Σ P 2 (see Theorem 5.10), and set B can be decided in Π P 2 (see Lemma 5.18). Therefore, deciding whether P has a majority optimum outcome is feasible in D P 2 .
For a lower bound of the problem Exists-Majority-Optimum we can exploit the construction used to show the lower bound of Is-Majority-Optimum, and obtain the following theorem, which proves that on mCP-nets deciding the existence of a majority optimum outcome is Π P 2 -hard.
Theorem 5.20. Let M be an mCP-net. Then, deciding whether M has a majority optimum outcome is Π P 2 -hard. Hardness holds even on polynomially connected classes of acyclic binary mCP-nets with indegree at most three and at most three agents.
By combining the two above results, we immediately conclude that deciding the existence of majority optimum outcomes over polynomially connected classes of acyclic mCP-nets is in D P 2 and Π P 2 -hard.
Corollary 5.21. Let C be a polynomially connected class of acyclic mCP-nets. Let M ∈ C be an mCP-net. Then, deciding whether M has a majority optimum outcome is Π P 2 -hard and in D P 2 .

Related work
In this section, we describe the larger context of this work in the literature, especially its relationship to previous work on compact representations of preferences, global and sequential voting, and the notion of O-legality.

Compact representations of preferences
The preferences of agents can be represented in different ways, and preference (representation) models can essentially be divided into quantitative and qualitative ones [8,45]. The former preference models associate with each outcome a numerical value, which is the value of a utility function, and preferences between outcomes are evaluated by comparing the utility values. The latter preference models provide a (not necessarily complete) order over the outcomes, which can be represented in multiple ways, e.g., via a plain sequence of outcomes, or, more generally, via a binary relation over the outcomes, i.e., a set of ordered pairs of outcomes, in which the first outcome of the pair is considered preferred to the second. The key point of qualitative preferences is that there is no precise quantification of the utility associated with the outcomes. For this reason, it has been argued in the literature that qualitative preference are easier to be stated by humans [8,12], as it is not necessary to estimate the utility values, which can be quite challenging. In this paper, we focus only on qualitative preference models, encoding preference relations. A preference model is extensive, if for any preference relation R that it is capable to represent, each outcome of the domain of R appears explicitly at least once in its representation of R. This implies that the size of its representation of R is at least linear in the number of outcomes of R. So, for example, a sequence of all the outcomes, or listing all the pairs of a preference relation, are both extensive representations. By definition, the number of possible outcomes in combinatorial domains is exponential in the number of features. Hence, the extensive representation of agents' preferences over combinatorial domains becomes quickly infeasible and unrealistic when the number of features is more than just a few. For this reason, compact formalisms to represent combinatorial preferences are needed [44,45,48]. An ideal compact representation scheme for combinatorial preferences would be one such that the space required for the representation is polynomial in the number of the features (and not in the number of the outcomes, like for extensive representations). Clearly, for information-theoretic reasons, it is not possible to have a preference model to compactly represent any possible preference relation over a combinatorial domain. Indeed, for a combinatorial domain characterized by n features of two values each, i.e., binary features, the number of all possible (complete) relations is (2 n Hence, no preference model could ever represent all the possible preference relations using only polynomial space in n [48]. Nonetheless, if we want to represent preference relations showing particular structures or patterns, then it is possible to take advantage of these patterns and "decompose" the relation to summarize it into a concise representation. In the literature, there has been considerable work on exploiting the structure of preferences to appropriately decompose them, and, e.g., Lang [45] gives a survey of different logical languages for compact preference representation (see also [8] for more references).
Among the preferences' structural patterns exploited to achieve compact representations, the (conditional) preference independence of the features is one of the most studied (see [8,18,42,45,46,47,48,75] and references therein). Intuitively, the (conditional) preferential independence of the features implies that the preference relation between outcomes varying on specific features can be influenced by the values of some features and can be totally independent from the values of some other features. When features are (conditionally) preferentially independent, it roughly means that portions of the structure of the preference relation are replicated throughout the preference relation. Therefore, these patterns in the preferences can be "factorised" to save space in the representation.
In the literature, (conditional) ceteris paribus preference statements have been proposed several times to compactly represent preferences with (conditional) preferential independencies among (sets of) features [8,45]. Moreover, a preference representation scheme should capture statements that are natural for agents to assess, and (conditional) ceteris paribus preference statements have several times been argued to be intuitive for users, as they resemble the way in which humans express their preferences and act upon them [7,8,37,38].
Ceteris paribus means "all else being equal", and ceteris paribus statements were classically defined to be non-conditional [37]. Given a set of features V ⊆ F, a non-conditional ceteris paribus preference statement sounds like: "Outcomes varying over V, and all else being equal, are ranked according to the following preference relation restricted over V". For example, a non-conditional ceteris paribus preference statement is "I prefer a round table to a square one, all else being equal" [37]. In this example, the set V contains the feature "shape" (of the table). Observe that, here, the "all else being equal" does not refer to the fact that the values of the other features determine the preference relation restricted over V. Instead, it means that the given ceteris paribus statement allows to compare outcomes varying only on V: outcomes varying over V and other features cannot be compared via the given statement.
Boutilier et al. [7,8] extended the idea of ceteris paribus statements to conditional ceteris paribus preference statements. In this case, given two disjoint sets of features V, Z ⊆ F, a conditional ceteris paribus preference statement sounds like: "Given the specific instantiation γ of values for the features in Z, outcomes varying over V, and all else being equal, are ranked according to the following preference relation restricted over V". For example, a conditional ceteris paribus preference statement could be "Given that the main course of the dinner is meat, I prefer a red wine to a white one, all else being equal". Here, Z contains the feature "main course", while V contains the feature "wine".
Among the representation schemes proposed in the literature based on (conditional) ceteris paribus statements, one using propositional logics was proposed by Lang [45], while Boutilier et al. [7,8] proposed CP-nets; for other representations based on (conditional) ceteris paribus statements, see references in [8,45]. More details on CP-nets are given in the introduction and in the preliminaries.

Global and sequential voting
The graphical structure of CP-nets shows that, in general, combinatorial preferences may exhibit dependencies between features. Dependencies are certainly a critical characteristic to model, especially to attain compact representations, however, they can become troublesome when combinatorial preferences are aggregated. Whether dependencies are actually problematic or not depends on the specific ways in which agents' votes are collected. These specific ways in which votes are collected are called voting protocols [19]. A voting protocol characterizes how a voting rule is implemented, i.e., it determines which information is elicited from the agents, and when this is done. In the literature, a number of different voting protocols for combinatorial vote have been considered (see, e.g., [18,43,48] and references therein). Two of them are global and sequential voting [18,43,46,47,48,75].
In global voting, agents' preferences over the entire combinatorial domain are collected at the same time, and then a voting rule is applied to the entire preference profile to select the winner(s). In case of (m)CP-nets, global voting consists in the agents communicating their whole CP-nets for vote aggregation. In sequential voting, votes are collected feature-by-feature. In particular, agents express at the same time their preferences over the individual features, and votes for the different features are collected sequentially in consecutive steps. In this case, a voting rule determines the winning value v for a feature, and, most importantly, agents are informed about v before the next feature is considered for voting in the protocol. Note that sometimes, in sequential voting, instead of voting for a sequence of single features, agents may be asked in a step of the protocol to express their preferences over the combined values (i.e., the value vectors) of a set of features (see, e.g., [1,75]). Global voting is the protocol of the voting semantics in mCP-nets.
Dependencies are not an issue in global voting, because agents communicate their preferences over the entire (exponential) combinatorial domain (or the entire CP-nets, if CP-nets are used to represent the agents' preferences), and hence all the information needed for the aggregation is available. However, global voting can be expensive to implement and evaluate, especially if extensive representations are adopted or preference relations are extensively unfolded from the compact representation before any further processing. Strictly speaking, from a theoretical perspective the computational complexity of aggregating preferences represented via extensive schemes can be lower than then complexity of performing the same task over compactly represented preferences. However, this computational simplification is artificial, because the computational complexity of a problem is evaluated relative to the size of the input. For extensive representations, the input is huge, and hence the computational complexity of problems over this kind of input can be low. Nevertheless, processing input of remarkable size can be computationally challenging. The computational burden of global voting can be limited by adopting sequential voting. A benefit of sequential voting over global voting is lowering the communication complexity, i.e., the amount of information needed to be exchanged by the agents to implement the protocol [18,19,48]. In sequential voting, agents are enquired in consecutive steps about their preferences for individual features, therefore the information required to be exchanged is only the preferences "projected" over individual features. However, feature dependencies can be quite detrimental for sequential voting, to the point that sub-optimal outcomes are selected-examples of this phenomenon are called multiple election paradoxes-or agents can experience regret after voting [18,43,46,47,48,75]. Intuitively, paradoxes may occur in sequential voting, because votes over the different features are collected separately, and this can clash with the different individual preferential dependencies that agents may have between features-more specifically, it might be the case that agents have to vote on a feature whose preferences depend on features for which it has not been voted yet [15].
Lacy and Niou [43] showed that these paradoxes in sequential voting can be (partly) avoided if the considered preferences are separable, i.e., they do not have dependencies among features. Intuitively, when represented via CP-nets, combinatorial preferences without dependencies among features do not have any edge between vertices. Clearly, this is a very strong assumption, and it is unlikely to be met in practice [46,47,75,76].

Overcoming multiple election paradoxes via O-legality
To overcome the strong limitation imposed by preference separability, Lang [46] proposed and investigated a weaker structural restriction of preferences, called O-legality, which preserves the nice properties of sequential voting (when evaluated over separable preferences) on a wider class of combinatorial preferences. Intuitively, a preference profile is O-legal, if the dependencies among the features for all the agents comply with a common sequence. More formally, if O = (F 1 , . . . , F m ) is a sequence of all the features of the combinatorial domain, a preference profile P is O-legal, if for any agent A and any two features F i and F j , if F i precedes F j in O, then A's preferences for F i do not depend on F j 's value [76]. The concept of O-legality has an immediate translation over CP-nets. Indeed, a profile of (preferences represented via) CP-nets is O-legal, if O is a topological order for all the CP-nets' graphs of the profile. Observe that the existence of a topological order for the features of a CP-net imposes that the graph of the CP-net is acyclic.
The intuitiveness of CP-nets to model combinatorial preferences, together with the convenient characterization of O-legality in CP-nets, has largely encouraged the study of sequential voting in O-legal (and acyclic) CP-nets. In a first group of works, the sequential composition of the voting rule assumed that a feature order was given beforehand (and the voting rule was defined upon the given order) [46,47,75]. Next, this idea was generalized by assuming that the sequential voting rule was not defined given the specific order [76], however, the existence of a shared topological order among the features was anyway required. A study on how well solutions computed via sequential voting approximate the winning outcomes obtained via global voting was presented in [20]. Also various other works considered O-legal CP-nets (see, e.g., [22,30,53,58,59]). Among them, an interesting approach to preference aggregation over O-legal CP-nets was proposed in [22], where "probabilistic" CP-nets were used to represent the result of the aggregation.

Going beyond O-legality
However, also O-legality is somewhat demanding, because it imposes that there are no "inversions" in the preference dependencies. For example, if in a profile of CP-nets encoding preferences for a dinner there were an agent whose choice of the main dish influences the choice of the wine and an other agent whose choice of the wine influences the choice of the main dish, then those CP-nets would not be O-legal. (Observe that the profile of CP-nets in Figure 2 is not O-legal). Hence, also assuming O-legality is in the end quite restrictive [51,68,77]. To go beyond the restrictions imposed by O-legality, different approaches were proposed.
One of these approaches is generalizing the idea of sequential voting. A voting agenda specifies the order in which the features have to be considered in sequential voting. If the voting agenda does not clash with any of the feature dependencies of all the agents, then it is possible to avoid multiple election paradoxes. In particular, if O is a shared topological order of the features of the CP-nets in a profile, then O is a voting agenda compatible with the feature dependencies of all the agents. A generalization of this idea is to have a sequence of elections for sets of joint features that cannot be decomposed due to preferential dependencies of some of the agents. In this way, the problem can be shifted to deciding suitable generalized voting agendas that do not clash with the agents' feature dependencies [1].
Another approach to overcome the limitations imposed by O-legality is hypercubewise preference aggregation [77]. This family of voting rules decomposes the preference aggregation task into two phases: first, a (hypercubewise) dominance graph is built by applying local voting rules to set of outcomes differing only for the value of a single specific feature (these outcomes are the neighboring vertices of the extended preference graph of CP-nets, which, in case of binary features, constitute a hypercube-this is where the name of the voting rule comes from); and then the winners are chosen from the hypercubewise dominance graph via choice sets functions, which may select dominating or undominated outcomes in the graph. The idea of the hypercubewise aggregation is at the base of the definition of the hypercubewise Condorcet winner (i.e., the hypercubewise majority winner, which is the outcome majority dominating all its neighbors) [21,77], also called the local Condercet winner, in which the hypercubewise dominance graph is obtained via majority voting. The definition of the hypercubewise Condorcet winner is different from the definition of the standard, or global, Condorcet winner (the majority optimum outcome, in this paper) obtained from global voting [21,77]. Also the dominance relation inferred from the hypercubewise dominance graph is different from the standard majority dominance relation obtained via global voting [51]. In [51], also other relations between local and global Condorcet winners are investigated. It was shown that deciding the existence of hypercubewise Condorcet winners is NP-complete [21,51].
A third approach proposes, similarly to the previous, to define new voting rules that can be applied also over profiles of non O-legal CP-nets. These new rules select the outcomes minimizing the total value of a loss function evaluated over the profile of CP-nets [67,68].

Analysis of global voting over CP-nets
Although the proposed approaches can deal with CP-net profiles that are not O-legal, they do not address the complexity analysis of global voting over (not necessarily O-legal) acyclic CP-nets. In fact, global voting over non-O-legal acyclic CP-nets has not received as much attention as sequential voting, although it was explicitly stated in the literature that a theoretical comparison between global and sequential voting would have been highly promising [46].
The first work studying global voting over (not necessarily O-legal) acyclic CP-nets was [64], in which mCP-nets were defined. Recall that the group dominance semantics of mCP-nets is global voting over a profile of CP-nets. Voting schemes over mCP-nets were considered from an algorithmic perspective in [64], which gave a computational insight for global voting over CP-nets. However, most of the algorithms considered in [64] were brute-force. Therefore, these algorithms gave only EXPTIME upper bounds for most of the global voting tasks over CP-nets, and no hardness result was provided in [64].
Algorithms exploiting SAT solvers to compute Pareto and majority optimal outcomes according to global voting over profiles of (not necessarily O-legal) acyclic CP-nets were proposed in [50] and [49], respectively. A SAT solver is used in [51] to compute, over profiles of even cyclic (and therefore also non O-legal) CP-nets, majority optimal and majority optimum outcomes according to global voting, starting from hypercubewise weak Condorcet winners. The approach of [51] was subsequently extended in [52] to consider also the possibility of multi-valued and incomplete CP-nets.
Despite the mentioned works advanced the study of global voting over (not necessarily O-legal) acyclic CP-nets, they still did not provide precise complexity results. As mentioned in the introduction, the precise complexity of these problems was actually reported as an open problem multiple times in the literature [46,49,50,51,52,67]. Our work is the first in the literature tackling directly the complexity analysis of dominance in mCP-nets (and hence the complexity of global voting over CP-nets).

Conclusion
In this paper, we have carried out a thorough complexity analysis of the Pareto and majority semantics in mCP-nets. Given the specific definitions of group dominance in mCP-nets, these results characterizes also the complexity of Pareto and majority global voting over CP-nets, which was missing and asked for in the literature various times. Unlike what is often assumed in the literature, in this work, we have not restricted the profiles of CP-nets to be O-legal, which makes the results achieved here more general. For the Pareto and the majority voting schemes, we have analyzed the problems of deciding dominance, optimal and optimum outcomes, and the existence of optimal and optimum outcomes. We have shown completeness results for most cases, which means that we have provided tight lower bounds for problems that (up to date) did not have any explicit lower bound transcending the obvious hardness due to the dominance test over the underlying CP-nets. Our hardness results are given for polynomially connected classes of binary acyclic (m)CP-nets. This means that our hardness results extend to classes of (m)CP-nets encompassing the CP-nets here considered, and in particular also to general mCP-nets with partial CP-nets or multi-valued features. The various problems analyzed here have been put at various levels of the polynomial hierarchy, and some of them are even tractable (in P or LOGSPACE), which is quite interesting given that for most of these tasks only EXPTIME upper-bounds were known in the literature.
There are various possible directions for further research. The lower bound for the problem of deciding the existence of majority optimum outcomes does not match the upper bound (Π P 2 -hardness and membership in D P 2 , respectively). Hence, it would be interesting to close this gap and find the precise complexity of the problem. Furthermore, characterizing the complexity of preference aggregation when partial CP-nets are allowed to be part of mCP-nets would be interesting, since with (standard) CP-nets, indifference between outcomes are not allowed. Having constraints on outcomes's feasibility is another interesting direction of investigation. Without any constraint, CP-nets model agents' preferences when it is assumed that all outcomes are attainable. However, this is not always the case. During the aggregation precess, we should take into account what outcomes are feasible. For example, to decide whether an outcome is majority dominated by another, we should check that the latter is actually feasible. It will be interesting studying the case in which constraints are issued over the outcome domain prior the preference aggregation and the case in which constraints are considered after the aggregation. A similar idea characterized the solution concepts in NTU cooperative games defined via constraints [32,33]. This approach could be merged with the definition of constrained CP-nets [9,62], Finally, it will also be interesting investigating structural restrictions on the structure of CP-nets, in the spirit of what was done in [13,31,34,35,39], to identify broader classes of CP-nets where the dominance test is tractable, whereas, in general, over acyclic CP-nets the dominance test is NP-hard.

A.1 Proofs for Section 3
Lemma 3.1. Let φ(X) be a Boolean formula in 3CNF defined over a set X of Boolean variables, and let σ X be an assignment on X. Let α σ X be the outcome of F (φ) encoding σ X on the feature set V, and assigning non-overlined values to all other features, and let β be the outcome assigning overlined values to all and only variable and clause features. Then: (1) There is an extension of σ X to X satisfying φ(X) if and only if β F(φ) α σ X .
(2) There is no extension of σ X to X satisfying φ(X) if and only if β F(φ) α σ X .
Proof. We first prove (1). (⇒) Assume that there is an extension σ X of σ X to X satisfying φ(X). To prove that β F(φ) α σ X we show that there is an improving flipping sequence from α σ X to β.
Recall that σ X is a complete assignment over X, and that if σ X is complete, then σ X = σ X . For each variable For all literals j,k evaluating to true in σ X , we flip the corresponding literal features P j,k from p j,k to p j,k .
Since σ X is a satisfying assignment, for each clause c j of φ, there is at least a literal j,k evaluating to true in σ X . For this reason, given any clause feature D j , at this point of the flipping sequence, there is at least one literal feature P j,k with value p j,k , and hence we can flip D j from d j to d j . We can do this for every clause feature.
Then, we flip to their overlined value all variable features that have not been flipped until now. By the definition of the CP tables of literal features, we can flip all features P j,k having value p j,k to p j,k , because in the outcome having been built so far through the flips shown above, for all pairs of features ( To conclude, observe that the obtained outcome is exactly β, and hence β F(φ) α σ X .
(⇐) Assume that β F(φ) α σ X . We show that there is an extension σ X of σ X to X satisfying φ(X). Since β F(φ) α σ X , there is an improving flipping sequence ρ : γ 0 → . . . → γ z from γ 0 = α σ X to γ z = β. Consider the truth assignment σ X built as follows: If there is an index q such that We first show that σ X is consistent, complete, and an extension of σ X to X. Observe that, by the definition of the CP tables of variable features, and the fact that those features have no parents, once a variable feature is flipped to its overlined value, it cannot be flipped back. Therefore, it cannot be the case that there are indices q and r for which there are variable features Hence, σ X is consistent. Moreover, we claim that, for any variable x i , there is always has a value in σ X , then the index q that we are looking for is q = 0 (because either Observe that this, along with the consistency of σ X proven above, implies that σ X [x i ] = σ X [x i ] for each variable x i having a truth value in σ X . On the other hand, if x i has not a value in σ X , then, since Hence, σ X is complete. To conclude, by all the properties above, σ X is also an extension of σ X to X. We now prove that σ X satisfies φ(X) by showing that σ X satisfies all clauses of φ(X).
Let c j be a clause of φ(X), and consider clause feature and γ t → Dj γ t+1 . Since flipping D j has to be an improving flip, it must be the case that there is a literal feature P j,k such that γ t [P j,k ] = p j,k . Therefore, since γ 0 [P j,k ] = p j,k , there is an index s < t such that γ s [P j,k ] = p j,k , γ s+1 [P j,k ] = p j,k , and γ s → P j,k γ s+1 . Now there are two cases: either (a) j,k = x i , or (b) j,k = ¬x i . For (a), since flipping P j,k has to be an improving flip, it must be the case that , and hence σ X satisfies c j . For (b), again since flipping P j,k has to be an improving flip, it must be the case that Therefore, σ X satisfies all clauses of φ, and hence σ X is an extension of σ X to X satisfying φ(X).
We now prove (2). We know that β F(φ) α σ X if and only if α σ X F(φ) β and β F(φ) α σ X . First, observe that α σ X F(φ) β is always true, as there is no improving flipping sequence from β to α σ X . Indeed, the values of variable features in β cannot be flipped to their non-overlined values according to their CP tables in F (φ), because they are the most preferred values and variable features do not have parents. So, β F(φ) α σ X if and only if β F(φ) α σ X . Hence, showing that there is no extension of σ X to X satisfying φ(X) if and only if β F(φ) α σ X is equivalent to showing that there is no extension of σ X to X satisfying φ(X) if and only if β F(φ) α σ X . However, we have already shown this in (1).

A.2 Proofs for Section 4
Lemma 4.5. Let φ(X) be a 3CNF Boolean formula, and let α be the outcome of M ipo (φ) assigning non-overlined values to all features. Then, φ(X) is satisfiable if and only if α is not Pareto optimal in M ipo (φ).
Proof. To prove the statement of the lemma, we first show the two following properties. Proof. Since φ is satisfiable, by inspection of the proof of Lemma 3.1, there is an improving flipping sequence in N ipo 1 that from α arrives to an outcome α 1 in which the values of all variable and clause features of F (φ) a are overlined. By definition of the interconnecting net H C (m), in N ipo 1 there is also an improving flipping sequence that from α 1 arrives to an outcome α 2 in which the values of all features (and also of the apex) of H C (m) are overlined. This allows α 2 to be further improved by a flipping sequence to an outcome α 3 in which the values of all variable and clause features of F (φ) b are overlined, because φ is satisfiable. Observe that α 3 = β.
Symmetrically, it can be shown that in N ipo 2 there exists an improving flipping sequence from α to the very same β. Therefore, β N ipo 1 α and β N ipo 2 α.
We now show that φ(X) is satisfiable if and only if α is not a Pareto optimal outcome of M ipo (φ). (⇒) Assume that φ(X) is satisfiable. Then, by Property 4.5.(2), there is an outcome β that is preferred to α by all agents of M ipo (φ). Therefore, β p Mipo(φ) α, and hence α is not Pareto optimal in M ipo (φ).
(⇐) Assume that φ(X) is not satisfiable. Then, by Property 4.5.(1), there is no outcome β that is preferred to α by all agents of M ipo (φ). Therefore, α is Pareto optimal in M ipo (φ).

A.3 Proofs for Section 5
Lemma 5.2. Let φ(X) be a Boolean formula in 3CNF defined over a set X of Boolean variables, and let σ X be an assignment on X. Let α σ X be the outcome of F s (φ) encoding σ X on the feature set V, and assigning non-overlined values to all other features. Let β be an outcome of F s (φ) such that β[U 1 U 2 ] = u 1 u 2 , assigning any value to the features of V, and assigning non-overlined values to all other features. Then: (1) There is an extension of σ X to X satisfying φ(X) if and only if β Fs(φ) α σ X ; improving flip, it must be the case that there is a literal feature P j,k such that γ q [P j,k ] = p j,k . Therefore, since γ 0 [P j,k ] = p j,k and γ q [P j,k ] = p j,k , there is an index p < q such that γ p [P j,k ] = p j,k , γ p+1 [P j,k ] = p j,k , and γ p → P j,k γ p+1 . By the definition of the CP table of P j,k , it must also be the case that p < s. This means that, for all the literal features that in ρ change their value from non-overlined to overlined, their flipping happens before the s-th step. Hence, if two different literal features linked to the same variable features V T i and V F i flip before the s-th step, then their flipping is based on consistent values assigned to V T i and V F i . Now there are two cases: either (a) j,k = x i , or (b) j,k = ¬x i . For (a), since flipping P j,k has to be an improving flip, it must be the case that , and hence σ X satisfies c j . For (b), again since flipping P j,k has to be an improving flip, it must be the case that Therefore, if β Fs(φ) α σ X , then there is an extension σ X of σ X to X satisfying φ(X).
We now prove (2). We know that β Fs(φ) α σ X if and only if α σ X Fs(φ) β and β Fs(φ) α σ X . First, observe that α σ X Fs(φ) β is always true, because there is no improving flipping sequence from β to α σ X , as the value u 1 of feature U 1 in β cannot be flipped in F s (φ), because it is the most preferred value of U 1 , and U 1 does not have parents. So, β Fs(φ) α σ X if and only if β Fs(φ) α σ X . Hence, showing that there is no extension of σ X to X satisfying φ(X) if and only if β Fs(φ) α σ X is equivalent to showing that there is no extension of σ X to X satisfying φ(X) if and only if β Fs(φ) α σ X . However, we have already shown this in (1). Proof. The intuition at the base of the proof is to put in relationship truth assignments over the variable set X with outcomes of M eml (Φ). In particular, given an assignment σ X over X, the associated outcome is β σ X , where σ X is encoded over the feature set V in the usual way, and all other features have non-overlined values. We show first that outcomes not in the form of a β σ X are not Majority optimal. Then, we show that if there is an assignment σ X such that (∀Y )¬φ(X/σ X , Y ) is valid, then β σ X is Majority optimal. On the other hand, if there is no assignment σ X for which (∀Y )¬φ(X/σ X , Y ) is valid, then none of the outcomes β σ X is Majority optimal.
To prove the statement of the lemma, we have to analyze the majority dominance relationships between outcome pairs. To organize this task, we define the following sets of outcomes: We show that only outcomes of a subset S (whose detailed characterization will be given toward the end of the proof) of O c might be Majority optimal. We do so by showing that (1) all outcomes in (O d ∪ O c ) \ S are majority dominated by some other outcome, and hence they are not Majority optimal; and that (2) all outcomes in S, which might be empty, are not majority dominated, and hence they are Majority optimal. Therefore, M eml (Φ) has a majority optimal outcome if and only if S is non-empty. We recall that, since M eml (Φ) is a 6CP-net, if α and β are two outcomes, |S M eml (Φ) (β, α)| ≥ 4 implies that β maj M eml (Φ) α. Property 5.11.(1). Let β ∈ O d be an outcome. Then, β is not Majority optimal in M eml (Φ).
Proof. There are two cases: either (1) β [U 1 U 2 ] = u 1 u 2 , or (2) β [U 1 U 2 ] = u 1 u 2 : (1) Let γ be the outcome assigning non-overlined values to all features in (V ∪ V ∪ W ∪ P ∪ D ∪ A ∪ B), and such that γ[U 1 U 2 ] = β [U 1 U 2 ]. We prove that γ maj M eml (Φ) β . Consider net N eml 3 . The following is an improving flipping sequence from β to γ, showing that γ N eml 3 β . We flip to their non-overlined value, in the following order, all features in V, V , W, P, D, and A, having an overlined value in β . Next, we flip in the proper order all features in B having an overlined value in β to their non-overlined value. The Proof. The intuition at the base of the proof is to put in relationship truth assignments over the variable set X with outcomes of M imm (Φ). In particular, given an assignment σ X over X, the associated outcome is β σ X , where σ X is encoded over the feature set V in the usual way, and all other features have non-overlined values. We show first that α majority dominates any other outcome β that is not in the form of a β σ X outcome, and hence none of them is Majority optimum. Moreover, for all such outcomes β σ X , β σ X does not majority dominates α, and hence, again, none of them is Majority optimum. So, only α is candidate to be Majority optimum. Then, we show that if there is an assignment σ X such that (∀Y )¬φ(X/σ X , Y ) is valid, then α maj Mimm(Φ) β σ X , and hence α is not Majority optimum, which implies that M imm (Φ) does not have any majority optimum outcome. On the other hand, if there is no assignment σ X for which (∀Y )¬φ(X/σ X , Y ) is valid, then, for all the outcomes β σ X , α maj Mimm(Φ) β σ X , which implies that α is majority optimum.
To prove the statement of the lemma we have to analyze the majority dominance relationship between α and the other outcomes. To organize this task, let us define the following sets of outcomes: