CONFLICT HYPERGRAPHS TO DEFINE NEW FAMILIES OF FACETS FOR THE INDEPENDENCE SYSTEM POLYTOPE

. In this paper, we develop a technique for exact simultaneous upliftings of circuit inequalities of an independence system polytope. The resulting inequalities define new families of valid inequalities for this polytope. They are obtained by simultaneously adding the most appropriate set of variables with the highest possible values of the lifting coefficient that maintain the validity. More specifically, in this technique, we introduce a procedure to generate two conflict hypergraph structures types: hypertrees and clutter. In this setting, we use the hyperedges cardinalities of these structures to compute the suitable lifting coefficient values. We then give necessary and sufficient conditions for both the circuit inequalities and the new families of valid inequalities to be facet-defining. We also give a condition of the positivity of the lifted circuit inequality coefficient values.


Introduction
An independence system on a finite set  = {1, . . ., } is a nonempty family ℐ of subsets of  closed under inclusion, i.e. ∀  ∈ ℐ, ∀  ⊂ ,  ⊂  ⇒  ∈ ℐ.The members of ℐ are called independent sets and those of 2  ∖ ℐ dependent sets, where 2  is the power set of  .Let  be a subset of  .Then  is a minimal dependent set, if  / ∈ ℐ and  ∖ {} ∈ ℐ for any  ∈ .A minimal dependent set is called a circuit.Similarly,  is a maximal independent set if  ∈ ℐ and  ∪ {} / ∈ ℐ for any  ∈  ∖  (see for example [26]).In a wide range of applications, a non-negative weight   is associated with each element ,  = 1, . . ., , of  .The Independence System Problem ISP consists of finding an independent set of maximum total weight.Formally, it can be formulated as follows: where  is a real -vector, and  ∈ ℐ means that there is  ∈ ℐ such that   = 1 if  ∈ , and   = 0 otherwise.Several combinatorial optimization problems, like the spanning tree problem, the stable set problem, the knapsack problem, and the matching problem are special cases of ISP.
The polytope associated with an ISP, denoted by , is the convex hull of the incidence vectors  of the independent sets in ℐ, i.e.:  = conv { ∈ {0, 1}  :  ∈ ℐ} .
Recall that, given a subset  of  , the rank of , denoted by (), is equal to the maximum cardinality of an independent set contained in  and the corresponding rank inequality, given by ∑︀ ∈   ≤ (), is valid for .For instance, the cover inequality for the knapsack polytope is a rank inequality.It is one of the most known valid inequalities for .In particular, when () = || − 1, the rank inequality is called a circuit inequality.
Theoretically, an ISP can be solved by using linear programming techniques if a complete description of  by a linear inequality system is known.This is, for instance, the case when the independence system is a matroid.Edmonds [7] showed that the matroid polytope is fully described by the rank inequalities and the non-negativity inequalities   ≥ 0, for all  ∈  .In practice, a partial description of  may be sufficient to solve efficiently the underlying ISP.Hence, finding some valid classes of facet-defining inequalities for  would be of great interest.
An independence system instance can be represented by using a conflict hypergraph  = (, ) by associating each element  of  to a vertex  of  , and associating a hyperedge for a set of vertices of  for which the corresponding elements in  conflict.Rank inequalities that are based on some structured subhypergraphs represent the most known facet-defining inequalities for .Conflict hypergraphs are used by numerous authors to generate both valid and facet-defining inequalities for .For instance, Euler et al. [8] generalized the clique inequalities to , and Fouilhoux et al. [9] derived a new family of facet-defining inequalities for .For more examples, the reader can refer to [9].In our work, we mainly describe two conflict hypergraph structures: hypertree and clutter, that we use to develop new families of valid inequalities for .
The rank inequality is a facet of a lower-dimensional polytope.Nevertheless, it can be lifted to define a facet for  by using lifting techniques.Lifting is a technique introduced by Gomory [10] which is used to strengthen valid inequalities.It also increases the coefficients of the variables while maintaining the validity of the inequalities.More specifically, in the lifting process, the variables can be lifted sequentially or simultaneously and the computation of the lifting coefficients can be either exact or approximate.In this framework, if the lifted variables are fixed at their lower bounds then the lifting is called uplifting, and if the lifted variables are fixed at their upper bounds, then the lifting is called down-lifting.Consequently, lifting can be conducted in at least eight ways depending on the combined properties of lifting: one from {up, down}, one from {exact, approximate}, and one from {sequential, simultaneous}.
In this work, we focus on the case of simultaneous lifting which is introduced by Zemel [30] in 1978.This technique is based on the search for the extreme points from the solutions of exponentially many linear integer programs.
More specifically, in this contribution, we develop a technique for exact simultaneous uplifting of circuit inequalities.Thereby, we aim to increase the dimension of the circuit inequality to construct new families of valid inequalities that we call CH inequalities.These inequalities are obtained by simultaneously adding a set of variables with the largest possible coefficient values that maintain the validity in .To this end, we introduce a procedure that we call CH procedure to generate two conflict hypergraph structures kinds: hypertree and clutter.Then, we use the hyperedges cardinalities of these conflict hypergraph structures to compute the lifting coefficient values of the added variables into the CH inequality.We describe a condition of the positivity of the lifted circuit inequality coefficient values.We also describe necessary and sufficient conditions for the circuit and CH inequalities to be facets-defining.
We have organized our paper as follows: Section 2 begins with some preliminaries and definitions of hypergraphs, ISP, and .In Section 3, we give a necessary and sufficient condition for the circuit inequality to be facet-defining.In Section 4, we state a condition of the positivity of the lifted circuit inequality coefficient values.We also present the steps of our lifting technique, describe the CH inequality and give the formula of its lifting coefficients.In Section 5, we describe the CH procedure and the resulting conflict hypergraph structures.In Section 6, we derive new families of valid inequalities for  by using the conflict hypergraph structures.In Sections 7 and 8, we give the theoretical results for hypertree and clutter structures, respectively, which provide necessary and sufficient conditions for the CH inequalities to be facets-defining.We end the paper in Section 9 with a conclusion and some directions for future research.

Hypergraphs
The concept of a hypergraph is an essential part of this work.It was introduced in 1970, with the work by Claude Berge [3].Formally, a hypergraph  is an ordered pair (, ) of finite sets, where  = {1, . . ., } is the set of vertices, and  = { 1 , . . .,   } is the set of hyperedges, where   ⊆  for all  = 1, . . ., .Note that a hypergraph whose all hyperedges have cardinality two is a graph.For background information on hypergraphs, the reader can refer to [4].In this concept, we concentrate on two structures: hypertree and clutter.Before defining these structures, we need to present some essential definitions.A realization of a given hypergraph  = (, ) is the undirected graph  = (, ) whose each edge is contained in a hyperedge of , and the subgraph of  induced by each hyperedge of  is connected.If  is a tree, it is called a tree-realization of .We now turn to define these structures: a hypertree is a hypergraph having a tree-realization [18], and a clutter is a hypergraph in which no hyperedge contains another one [11].

ISP and 𝒫
This article aims to define valid inequalities that could be facets for .In this context, and more specifically, in certain proofs developed in the following sections, we first need to know the dimension of  to guide the construction of the facet proof.Before computing the dimension of , let us first describe how we can write ISP and  in matrix forms.
where  ∈ R × + is a matrix of resource requirements, and  ∈ R  + is a vector of resource capacities.Without loss of generality, we assume that   ≤   , for  ∈ {1, . . ., } and  ∈ {1, . . ., } in every ISP instance.Hence, every dependent set has at least two elements.In this case,  is of full dimension (i.e.dim() = ), since the ( + 1) vectors { 1 , . . .,   , 0}, where   denotes the j th unit vector (the vector with 1 in the j th component, and 0's elsewhere), are affinely independent points.The facets associated with the inequalities   ≥ 0,  = 1, . . .,  are known as trivial facets.
Remark 1.As dim() = , it is clear that, in the facet proof of the sufficient condition of the CH inequality, we should exhibit  affinely independent points that fulfill this inequality with equality.
Consequently, to avoid repetition, we only have to list the  affinely independent points without repeating Remark 1 in each proof.
Alternatively, as an independence system is fully characterized by its family of circuits, then one can describe  by (see [20]): where C denotes the family of circuits of the ISP associated with the polytope .

Examples of ISP
In combinatorial optimization, many problems can be considered in terms of maximizing an objective function over an independence system.We cite below some examples of ISP.
The stable set problem consists of finding in a vertex weighted given graph a stable set (a subset of pairwise non-adjacent vertices) of maximum weight.It is an ISP, where a stable set represents an independent set.
The triangle-free induced subgraph problem (TFISP) consists of finding, in a vertex weighted given graph , a triangle-free induced subgraph whose weight is maximum [9].The feasible solutions of TFISP are the independents of the independence system, where each triangle-free induced subgraph of  represents an independent set.
The knapsack problem is also an ISP, for which the associated independence system is given as follows: the set  is the set of items, and the set ℐ is the family of subsets of  whose incidence vectors satisfy the knapsack constraint ∑︀ ∈     ≤  with   > 0 for all  ∈  , where  is the capacity of the knapsack and   is the weight of the item  of  .The knapsack problem consists of finding an independent set with maximum profit, a positive profit being assigned to each item.

Circuit inequality
The rank inequality is among the most crutial classes of valid inequalities for .For any independence system (, ℐ) and any subset  of  , the inequality which is called rank inequality, is valid for , where () denotes the rank of  in (, ℐ).It is determined by using the rank set function  defined as follows: where The rank (), of a subset  represents the maximum cardinality of an independent set in (, ℐ) that is containd in .
When the set  is a circuit, we have () = || − 1, and the rank inequality is called circuit inequality.Sufficient conditions for the rank inequality to define a facet for  are given by Laurent [20].In this section we present a necessary and sufficient condition for the circuit inequality to define a facet for .
Theorem 1.Let us consider an independence system (, ℐ), and a subset  of  , where  is a circuit.The circuit inequality: induced by  defines a facet for  if and only if the following condition holds: Proof.Necessary condition.
For the sake of contradiction, assume that the condition () of Theorem 1 is not satisfied.If there exists  ∈  ∖  such that ∀  ∈  the set   = ( ∖ {}) ∪ {} is a circuit, then any solution of the independent set problem containing  can not include more than || − 2 elements from .Hence, its incidence vector cannot satisfy Inequality (8) at equality.It follows that  cannot belong to any solution of the problem whose incidence vector satisfies Inequality (8) at equality.We then deduce that Inequality ( 8) is equivalent to   ≥ 0. Since Inequality ( 8) is not a positive multiple of   ≥ 0, this implies that it does not define a facet for .

Sufficient condition
We know that a circuit inequality is valid.The  following feasible points are affinely independent and satisfy Inequality (8) at equality.
Note that these  points are affinely independent.Thus, the circuit inequality define de facet for .

Lifted circuit inequality
The circuit inequality induced by a subset  of  : is a facet of the lower dimensional polytope . Nevertheless, we can lift it to define a facet for  using lifting techniques, where the variables   ,  ∈  ∖ , could be introduced into the inequality by finding appropriate lifting coefficients.
In this work, we focus our interest on the lifted circuit inequality of the form: where  =  ∖ , obtained from introducing the variables   ,  ∈  ∖  into (9) with coefficients   ≥ 0.
In this context, we will develop new families of valid inequalities using conflict hypergraph structures that we call CH inequality (conflict hypergraph inequality).Before presenting the CH inequality, let's first answer the following question: Could the variables   ,  ∈  ⊆  ∖ , be lifted into the circuit inequality?
To guarantee the lifting of the variables   ,  ∈  ∖ , the coefficients   should be strictly positive.In Corollary 1, we state a condition of the positivity of the coefficients   in Inequality (10).This condition is based on the conflict relationship between every two elements  and  in an independence system instance on  , where  belongs to a circuit  and  belongs to  ∖ .To represent this conflict relationship for any circuit  ⊂  , we associate a critical bipartite graph   = (︀ , S ,  )︀ , with S ⊆  ∖ , to an independence system instance on  as follows: -The sets  and S are the bipartition of the vertex set, and each vertex  in   is associated with a binary variable   .-The set  is the edge set.An edge {, } ∈  links vertices  and , where  ∈  and  ∈ S , if and only if , associated to an independence system on  , is complete bipartite, then we can find   > 0, ∀  ∈ S , such that Inequality (10) holds.
Proof.As the critical bipartite graph   = (︀ , S ,  )︀ is complete bipartite, then the set   =  ∖ {}) ∪ {} is a circuit, ∀ {, } ∈  such that  ∈  and  ∈ S .It follows that   is not a feasible solution for ISP.
For the sake of clarity, in the rest of this proof, Inequation (10) will be written in the form  ≤ , where

So, we have
Since  is a circuit,  ∖ {} is a feasible solution for ISP.It follows that; From (A) and (B), we get: Hence ∀  ∈ S ,   > 0.
We now turn to the development of CH inequality.To this end, we will introduce a technique for an exact simultaneous uplifting of the circuit inequality based on conflict hypergraph structures, which are used to compute the lifting coefficient values.This technique consists of the following steps: (1) Step 1: Define the CH inequality.(2) Step 2: Introduce a procedure to generate two conflict hypergraph structures kinds: hypertree and clutter.
(3) Step 3: Use the generated conflict hypergraph structures to compute the lifting coefficient of CH inequality.
Steps 2 and 3 will be presented in detail in Sections 5 and 6, respectively.This section is devoted to Step 1, which consists of defining the CH inequality by describing its form and giving the formula of its lifting coefficients.
Let us start with the form of the CH inequality, which we define as follows: where  ⊆  ∖ .Inequality (11) can be seen as a special case of Inequality (10), where the coefficients   are all equal.The difference between them is in the way in which the variables   are introduced into the circuit inequality, sequentially for Inequality (10), and simultaneously for Inequality (11).
At the end of this section, we present how the value of the lifting coefficient  could be computed.Before starting, let us first introduce some notations.Given a circuit  of ISP, where  =  ∪,  ⊂ , 0 ≤ || ≤ ||−1,  ⊂  and 2 ≤ || ≤ | | − 1, one can observe that  ∖ {}, where  ∈ , is a feasible solution of ISP.We can compute the lifting coefficient  by using the following formula: where . To make Formula (12) simpler, let us set  =  ∖ , || =  + 1 and let  = ||, where  is the number of elements of  that do not belong to , and  is the number of elements of  that we can add to  to form a maximal independent.It follows that || = || − , and thus we can use the following formula to compute the lifting coefficients :

Conflict hypergraph structures
In this section, we introduce a procedure, which we call CH procedure (conflict hypergraph procedure), to generate two conflict hypergraph structures kinds: hypertree and clutter.This procedure allows forming a conflict hypergraph, denoted by   = (  ,   ), for a given independence system instance ISP, on a finite set  of  elements and a given value of the parameter .The set  defines the set of vertices   .

Begin
Step 0 Initialization: Step 1 Select from   elements forming a circuit and put them in  0 .
Step 2 If  0 is empty, then go to Step 9.
Step 7 Select a subset of elements from  () , namely  () , to add to   in order to form a circuit, such that | () | ≥ 2.
Step 9 Stop and do the following test: If  0 ̸ = ∅ and   ∖  0 ̸ = ∅, then sort the elements of the set   ∖  0 in increasing order of their cardinal.Then according to this sort, re-index sets   , for  = 1, . . ., , from the largest one to the smallest one with index  from  to 1, i.e.   = { 0 ,   , . . .,   , . . .,  1 }.Thereafter, re-index the elements of the set   .It follows that: - where   = ∑︀ −1 =1 ( () + 1) and 1 ≤  () ≤  (−1) .Else write "  = (  ,   ) could not be constructed using these data".End From this procedure, we can derive a conflict hypergraph   = (  ,   ) with a set of vertices   =  , and a set of hyperedges   = { 0 ,   , . . .,  1 } which are circuits.According to the value of parameter , we can distinguish two structures of   = (  ,   ): -Hypertree structure: We suppose that there exist two oracles, the first one checks in unitary time if a given set is a circuit, and the second one checks in unitary time the condition () of Theorem 1.Thus, under this hypothesis, we can deduce that the running time of the CH procedure in the worst case is quadratic ( 2 ).So, the conflict hypergraphs   = (  ,   ) can be generated in polynomial time.
In the following sections, the conflict hypergraph   = (  ,   ) will be referred to as a conflict hypergraph associated with an independence system instance.
Remark 2. Let  = ( 1 , . . .,   ) be the weight vector associated with the elements of  .For problems with capacity constraints, like the knapsack problem, it is recommended to sort the elements of  in decreasing order according to their weight.This allows us to easily have the hypergraph structures.

New families of valid inequalities for 𝒫
In this section, we present the use of the CH procedure to simultaneously exact uplifting the circuit inequalities into CH inequalities.Indeed, to compute the lifting coefficient , we use hyperedge cardinalities of the conflict hypergraph   = (  ,   ) to define the values of the parameters || and | () |, where  () =   ∖ .The CH inequalities induced by a circuit  0 of   = (  ,   ) are given by: where As the hyperedges   ,  = 1, . . ., , are circuits, it follows that   ∖ {}, where  ∈  () , is a feasible solution of ISP.Thus, by Formula ( 12), when  =  0 , the lifting coefficients  () are given by  () = −1  () .
Remark 3. The values of the coefficients  () vary according to the cardinalities of the hyperedges  0 and   ,  = 1, . . ., .We can distinguish two cases: -Case 1: -Case 2: The CH inequalities represent new families of valid inequalities for .We state the validity of the CH inequalities in Theorem 2. Throughout this paper, we use the following notations to ease the proofs of the theorems.z1 = ∑︁ ∈    ∖{}  ≤ || and z2 = ∑︁ Theorem 2. Let us consider an independence system instance with the corresponding conflict hypergraph   = (  ,   ).Then Inequality ( 14) is valid for .
Remark 4. For  = 1, when the elements of the vertex set   are arranged in increasing order, according to their weight, Inequalities (14) reduce to the extended cover inequality.
The CH procedure allows for forming a conflict hypergraph   = (  ,   ).The set of hyperedges   could contain only one hyperedge, namely  0 .Nevertheless, this does not avoid generating valid inequalities for .Indeed, from this kind of conflict hypergraphs, we can generate CH inequalities of the form In the following theorem, we state the validity of Inequality (30).
Proof.Assume for contradiction that Inequality (30) is not valid.Thus, there exists  * ∈  such that where  1 ∈  (1) .Since  0 is a circuit, we necessarily have z1 ≤ | 0 | − 1.Thus, we can distinguish the following cases: -Case 1.If   1 = 0, then Inequality (31) reduces to Inequality (32) which means that the circuit inequality induced by the circuit  0 is not valid for , which is absurd.-Case 2. If   1 = 1, then Inequality (31) In this case we can distinguish two subcases: In the next two sections, we will focus on obtaining necessary and sufficient conditions for CH inequalities to be facets defining for .The first condition consist of the positivity of the lifting coefficient .Indeed, the critical bipartite graph, associated to an independence system instance on  , for which the sets  0 and  ()  are the bipartition of the vertex set, should be complete bipartite.
Remark 5.This condition arise in each of the Theorems 4-8, and it proof could be done in the same way as in Corollary 1.
In the remaining of this paper, and for the sake of ease and clarity of the theorem proofs, let us note by ℱ () , where a face of  defined by Inequality (14).And let us also note the sets  () and ℱ () by  (1) and ℱ (1) , respectively, when  = 1, otherwise, when  > 1, we let them unchanged.Moreover, to prove that Inequality ( 14) is facet defining for , it suffices to show that ℱ () contains  affinely independent feasible points.

Theoretical results for the hypertree structure
The lifting technique presented in the previous sections allows extending a circuit inequality into a CH inequality.However, it does not guarantee that the generated CH inequalities define facets for .In this section, we present, in Theorems 4-6, respectively, necessary and sufficient conditions for CH inequalities to be facet-defining for , for the hypertree structure.Theorem 4. Let us consider an independence system instance with the corresponding conflict hypergraph   = (  ,   ).For  = 1, Inequality ( 14) is a facet-defining for  if and only if the following conditions are satisfied: (i) The critical bipartite graph  0 = (︀  0 ,  (1) ,  )︀ is complete bipartite.(ii)  2 = {︀ 3, . . ., | 0 |,  −  (1) + 1, . . .,  }︀ , is not a circuit.(iii) There exist  1 ,  2 ∈  1 ∖  and ℎ 3 ∈  (2) such that the set Proof.Necessary condition.

Theoretical results for the clutter structure
In this section we present necessary and sufficient conditions for the CH inequalities to be facet-defining for  for the clutter structure.
Let us consider Inequality (35).Assuming conversely that Inequality (14) for  = 1 does not satisfy the conditions of Theorem 7 and defines a facet for .
Let us consider Inequality (36).Assuming for contradiction that Inequality (14) for  > 1 does not satisfy the conditions of Theorem 8 and defines a facet for .
-From the conditions () and () and as the set   is a circuit, we obtain )︁⃒ ⃒ ⃒.Clearly, these feasible points are affinely independent.This establishes that Inequality (14) for  > 1 defines a facet for .
And hence by Theorem 7, it follows that the inequality defines a facet for .

Conclusion
In this paper, we have presented a technique for an exact simultaneous uplifting of circuit inequalities.In this technique, we used conflict hypergraph structures to define new families of valid inequalities for the independence system polytope.These inequalities are obtained by simultaneously adding the most appropriate set of variables with the largest coefficient values that maintain their validity.Furthermore, we have introduced a procedure to generate two conflict hypergraph structures kinds: hypertrees and clutters.Moreover, we used the hyperedges cardinalities of these structures to compute the lifting coefficient values of CH inequalities.
On the other hand, we have provided some theoretical results that consist of necessary and sufficient conditions for the circuit and the CH inequalities to be facet-defining for .We also give a condition of the positivity of the lifted circuit inequality coefficient values.
As perspectives, it would be interesting to test our inequalities in the separation process of branch-and-cut algorithms to investigate their computational potential in solving ISP.
Beside, investigating other structures of conflict hypergraphs and selecting elements of the independence system would be of great interest.This could serve us to generate more new families of facets defining inequalities for a known ISP.

Figure 1 .
Figure 1.Examples of a hypergraph and a clutter.