FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

We study the \textsc{$\alpha$-Fixed Cardinality Graph Partitioning ($\alpha$-FCGP)} problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph $G$, two numbers $k,p$ and $0\leq\alpha\leq 1$, the question is whether there is a set $S\subseteq V$ of size $k$ with a specified coverage function $cov_{\alpha}(S)$ at least $p$ (or at most $p$ for the minimization version). The coverage function $cov_{\alpha}(\cdot)$ counts edges with exactly one endpoint in $S$ with weight $\alpha$ and edges with both endpoints in $S$ with weight $1 - \alpha$. $\alpha$-FCGP generalizes a number of fundamental graph problems such as \textsc{Densest $k$-Subgraph}, \textsc{Max $k$-Vertex Cover}, and \textsc{Max $(k,n-k)$-Cut}. A natural question in the study of $\alpha$-FCGP is whether the algorithmic results known for its special cases, like \textsc{Max $k$-Vertex Cover}, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for \textsc{Max $k$-Vertex Cover} is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for $\alpha>0$ and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with $\alpha>1/3$ and minimization with $\alpha<1/3$.


Introduction
In this work, we study a broad class of problems called α-Fixed Cardinality Graph Partitioning (α-FCGP), originally introduced by Bonnet et al. [2] 1 .The input is a graph G = (V, E), two non-negative integers k, p, and a real number 0 ≤ α ≤ 1.The question is whether there is a set S ⊆ V of size exactly k with cov α (S) ≥ p (cov α (S) ≤ p for the minimization variant), where Here, m(S) is the number of edges with both endpoints in S, and m(S, V \ S) is the number of edges with one endpoint in S and other in V \ S. We will call the maximization and minimization problems Max α-FCGP and Min α-FCGP, respectively.This problem generalizes many problems, namely, Densest k-Subgraph (for α = 0), Max k-Vertex Cover2 (for α = 1/2), Max (k, n − k)-Cut (for α = 1), and their minimization counterparts.
Although there are plethora of publications that study these special cases, the general α-FCGP has not received much attention, except for the work of Bonnet et al. [2], Koana et al. [19], and Schachnai and Zehavi [23].In this paper, we aim to demonstrate the wider potential of the existing algorithms designed for specific cases, such as Max k-Vertex Cover, by presenting an algorithm that can handle the more general problem of α-FCGP.Algorithms for these specific cases often rely on greedy vertex degree orderings.For instance, Manurangsi [20], showing that a (1−ε)-approximate solution can be found in the set of O(k/ε) vertices with the largest degrees, gave a (1 − ε)-approximation algorithm for Max k-Vertex Cover that runs in time (1/ε) O(k) • n O (1) .Fomin et al. [14] gave a 2 O( √ k) • n O(1) -time algorithm for Max k-Vertex Cover on apex-minor graphs via bidimensionality arguments, by showing that an optimal solution S is adjacent to every vertex of degree at least d + 1, where d is the minimum degree over vertices in S. In this work, we will give approximation algorithms as well as subexponential-time algorithms for apex-minor free graphs exploiting the greedy vertex ordering.
Next, we discuss the regime of subexponential-time algorithms.Amini et al. [1] showed that Max k-Vertex Cover is FPT on graphs of bounded degeneracy, including planar graphs, giving a k O(k) • n O(1) -time algorithm.They left it open whether it can be solved in time 2 o(k) • n O (1) .This was answered in the affirmative by Fomin et al. [14], who showed that Max k-Vertex Cover on apex-minor free graphs can be solved in time -time algorithm for Max α-FCGP with α > 1/3 and Min α-FCGP with α < 1/3.The complexity landscape of Max α-FCGP with α < 1/3 (and Min α-FCGP with α > 1/3) is not well understood.It is a long-standing open question whether Densest k-Subgraph on planar graphs is NP-hard [4].Note that the special case Clique is trivially polynomial-time solvable on planar graphs because a clique on 5 vertices does not admit a planar embedding.
We use the standard graph-theoretic notation and refer to the textbook of Diestel [10] for undefined notions.In this work, we assume that all graphs are simple and undirected.For a graph G and a vertex set S, let G[S] be the subgraph of G induced by X.For a vertex v in G, let d(v) be its degree, i.e., the number of its neighbors.For vertex sets X, Y , let m(X) In this work, an optimal solution for Max α-FCGP (and Min α-FCGP) is a vertex set S of size k such that cov α (S) ≥ cov α (S ′ ) (resp., cov α (S) ≤ cov α (S ′ )) for every vertex set of size k.A graph H is a minor of G if a graph isomorphic to H can be obtained from G by vertex and edge removals and edge contractions.Given a graph H, a family of graph H is said to be H-minor free if there is no G ∈ H having H as a minor.A graph H is an apex graph if H can be made planar by the removal of a single vertex.
We refer to the textbook of Cygan et al. [5] for an introduction to Parameterized Complexity and we refer to the paper of Marx [22] for an introduction to the area of parameterized approximation.

FPT Approximation Algorithms
In this section, we design an FPT Approximation Scheme for Max α-FCGP as well as Min α-FCGP parameterized by k and α, assuming α > 0.More specifically, we prove the following theorem.
▶ Theorem 1.For any 0 < α ≤ 1 and 0 < ϵ ≤ 1, Max α-FCGP and Min α-FCGP each admits an FPT-AS parameterized by k, ϵ and α.More specifically, given a graph G = (V, E) and an integer k, there exists an algorithm that runs in time f (k, α, ϵ) • n O (1) , and finds a set for Min α-FCGP, where O ⊆ V is an optimal solution.
For the case that OPT := cov α (O) is large, the following greedy argument will be helpful.

M F C
To this end, we note that O i is obtained from O i−1 by removing w i and adding y i .Thus, where Lemma 2 allows us to find an approximate solution when OPT is sufficiently large.The case that OPT is small remains.We use different approaches for Max α-FCGP and Min α-FCGP.

Algorithm for MAX α-FCGP
Let v 1 be a vertex with the largest degree.Our algorithm considers two cases depending on whether d(v 1 ) > ∆ := 2k 2 ϵα + k.If d(v 1 ) > ∆, we can argue that the set S from Lemma 2 a (1 − ϵ)-approximate solution.To that end, we make the following observation.
where the inequality follows from the fact that at most k edges incident to v 1 can have the other endpoint in S.This implies that Where we use the assumptions that 0 < α ≤ 1 and d(v 1 ) ≥ ∆. ◀ Thus, for d(v 1 ) > ∆, we have OPT ≤ cov α (S) + 2k 2 ≤ (1 + ε) • cov α (S), and thus cov α (S) ≥ (1 − ε) • OPT.So assume that d(v 1 ) < ∆.In this case, the maximum degree of the graph is bounded by ∆.Let O ⊆ V be an optimal solution.Then the total number of edges contributing to cov α (O) is bounded by k∆ = O(k 3 /αϵ).Let Q be the set of vertices in V \ O that have a neighbor in O, and note that |Q| = O(k 3 /αϵ).Let z = |O| + |Q|, and note that z = O(k 3 /αϵ).
We first guess the structure of the subgraph , where E ′ consists of all edges with at least one endpoint in O.For each guess for G ′ , we check whether there exists a subgraph in G that is isomorphic to G ′ .Over all guesses where we find an isomorphic subgraph, we return the solution maximizing the cov α (•) value.Note that the number of guesses is bounded by 2 z 2 = g(k, α, ϵ).Since the maximum degree of G is bounded by ∆, and the number of vertices in the subgraph corresponding to each guess is z, we can solve each instance of Subgraph Isomophism in time 2 O(z∆) using random separation, e.g., Theorem 5.7 in [5].Thus, overall, the running time of the algorithm is bounded by some f (k, α, ϵ) • n O (1) .Combining both cases, we conclude the proof of Theorem 1.

Algorithm for MIN α-FCGP
For Min α-FCGP, our algorithm considers two cases depending on the value of OPT.If OPT ≥ 2k 2 ε , then our algorithm returns the set S from Lemma 2. Note that cov α (S) Now suppose that OPT < 2k 2 ε .In this case, we know that O cannot contain a vertex of degree larger than ∆ = 2k 2 αϵ + k, for otherwise, cov α (O) > α(∆ − k) ≥ OPT, which is a contradiction.
In this case, we can guess the structure of , where E ′ consists of all edges with at least one endpoint in E ′ .Then, we can find a subgraph isomorphic to G ′ using an FPT algorithm (we can delete the edges between all vertices whose degree is larger than ∆).This takes FPT time.
Since the value of OPT is unknown to us, we cannot directly conclude which case is applicable.So we find a solution for each case and return a better one.

Subexponential FPT Algorithm for MAX α-FCGP on Apex-Minor Free Graphs
Fomin et al. [14] showed that Partial Vertex Cover on apex-minor free graphs can be solved in time 2 1) .In this section, we will prove its generalization to Max α-FCGP as well as Min α-FCGP: ▶ Theorem 4. For an apex graph H, let H be a family of H-minor free graphs.
We will give a proof for the maximization variant.The minimization variant follows analogously.Let σ = v 1 , v 2 , . . ., v n be an ordering of vertices of V in the non-increasing order of degrees, with ties broken arbitrarily.That is, We will denote the graph by G = (V σ , E) to emphasize the fact that the vertex set is ordered w.r.t.σ.We also let V j σ = {v 1 , . . ., v j }.We first prove the following lemma.Let C = {u i1 , u i2 , . . ., u i k } be the lexicographically smallest solution for Max α-FCGP and Proof.Suppose for the contradiction that C is not a dominating set for G[V j σ ].Then, there exists a vertex v i with 1 . Define the following: We will show that C ′ is another solution for the Max α-FCGP instance.Since C ′ \ {v i } = C \ {v j }, it suffices to show that where the inequality is due to the assumption that α ≥ 1/3.Therefore, which is a contradiction to the assumption that C is the lexicographically smallest solution for Max α-FCGP.◀ In view of Lemma 5, we can use the following approach to search for the lexicographically smallest solution C. First, we guess the last vertex v j of C in the ordering σ, i.e., we search for a solution C such that v j ∈ C and C ⊆ V j σ .If G[V j σ ] has no dominating set of size at most, say 2k, then we reject.This can be done in polynomial time, since Dominating Set admits a PTAS on apex-minor free graphs [7].We thus may assume that there is a dominating set of size 2k in G[V j σ ].It is known that an apex-minor free graph with a dominating set of size κ has treewidth O( √ κ), where O hides a factor depending on the apex graph whose minors are excluded [6,9,12].We can use a constant-factor approximation algorithm of Demaine [8] to find a tree decomposition T of width w ∈ O( √ k).Finally, we solve the problem via dynamic programming over the tree decomposition.Bonnet et al. [2] gave a O * (2 w )-time algorithm that solves Max α-FCGP with a tree decomposition of width w given.We need to solve a slightly more general problem because T is the tree decomposition is over V j σ .To remove V \ V j σ , we introduce a weight ω : For Min α-FCGP, we can show the following lemma whose proof is omitted because it is almost analogous to the previous one.The only change is that, V σ refers to the vertices in the non-decreasing order of degrees.Also, we consider the regime where 0 ≤ α ≤ 1/3, which implies α ≤ 1 − 2α, which would give the reverse inequality in (1).▶ Lemma 6.Let G = (V σ , E) be a yes-instance for Max α-FCGP, where 0 ≤ α ≤ 1/3.Let C = {u i1 , u i2 , . . ., u i k } be the lexicographically smallest solution for Max α-FCGP and u i k = v j for some j.Then C is a dominating set of size k for G[V j σ ].

Conclusion
In this paper, we demonstrated that the algorithms exploiting the "degree-sequence" that have been successful for designing algorithms for Max k-Vertex Cover naturally generalize to Max/Min α-FCGP.Specifically, we designed FPT approximations for Max/Min α-FCGP parameterized by k, α, and ϵ, for any α ∈ (0, 1].For Max α-FCGP, this result is tight since, when α = 0, the problem is equivalent to Densest k-Subgraph, which is hard to approximate in FPT time [21].We also designed subexponential FPT algorithms for Max α-FCGP (resp.Min α-FCGP) for the range α ≥ 1/3 (resp.α ≤ 1/3) on any apex-minor closed family of graphs.It is a natural open question whether one can obtain subexponential FPT algorithms for Max/Min α-FCGP for the entire range α ∈ [0, 1].A notable special case is that of Densest k-Subgraph on planar graphs.In this case, the problem is not even known to be NP-hard, if the subgraph is allowed to be disconnected.For the Densest Connected k-Subgraph problem, it was shown by Keil and Brecht [18] that the problem is NP-complete on planar graphs.From the other side, it can be shown that Densest Connected k-Subgraph admits a subexponential in k randomized algorithm on apex-minor free graphs using the general results of Fomin et al. [13].Thus, dealing with disconnected dense subgraphs is difficult for both algorithms and lower bounds.
The objective is then to maximize cov α (C) + α v∈C ω(C).The dynamic programming algorithm of Bonnet et al. can be adapted to solve this weighted variant in the same running time.Thus, we obtain a 2 O( √ k) • n O(1) -time algorithm for Max α-FCGP.

S 2 0 2 3 46:4 FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges
For Max α-FCGP, let S be the set of k vertices with the largest degrees.Then, cov α (S) ≥ OPT − 2k 2 .For Min α-FCGP, let S be the set of k vertices with the smallest degrees.Then, cov α (S) ≤ OPT + 2k 2 .Here, we index the vertices so that d(y i ) ≥ d(y j ) and d(w i ) ≥ d(w j ) (for Min α-FCGP, d(y i ) ≤ d(y j ) and d(w i ) ≤ d(w j )) for i < j.Note that due to the choice of S, it holds that d(yi ) ≥ d(w i ) (d(y i ) ≤ d(w i ) for Min α-FCGP) for each 1 ≤ i ≤ t.Now we define a sequence of solutions O 0 , O 1 , . . ., O t , where O 0 = O, and for each 1 ▶ Lemma 2.Proof.Without loss of generality, we assume that O ̸ = S. Let S \ O = {y 1 , y 2 , . . ., y t }, and O \ S = {w 1 , w 2 , . . ., w t }, where 1 ≤ t ≤ k.