On the (Parameterized) Complexity of Recognizing Well-covered (r,l)-graphs

An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓℓ cliques. A graph is (r,ℓ)(r,ℓ) if it admits an (r,ℓ)(r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)(r,ℓ)-well-covered if it is both (r,ℓ)(r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)(r,ℓ)-Well-Covered Graph problem ((r,ℓ)(r,ℓ) wcg for short), we are given a graph G, and the question is whether G is an (r,ℓ)(r,ℓ)-well-covered graph. In the Well-Covered (r,ℓ)(r,ℓ)-Graph problem (wc (r,ℓ)(r,ℓ) g for short), we are given an (r,ℓ)(r,ℓ)-graph G together with an (r,ℓ)(r,ℓ)-partition of V(G) into r independent sets and ℓℓ cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for r≥3r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size αα of a maximum independent set of the input graph, its neighborhood diversity, or the number ℓℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by αα can be reduced to the wc (0,ℓ)(0,ℓ) g problem parameterized by ℓℓ, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP⊆NP/polycoNP⊆NP/poly

Hitting forbidden induced subgraphs on bounded treewidth graphs

International audienceFor a fixed graph H, the HIS Deletion problem asks, given a graph G, for the minimum size of a set S ⊆ V (G) such that G \ S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function fH (t) such that HIS Deletion can be solved in time fH (t) • n O(1) assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that fH (t) = 2 O(t h−2) for every graph H on h ≥ 3 vertices, and that fH (t) = 2 O(t) if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [Inf. Comput. 2017] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function fH (t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then fH (t) = 2 Θ(t h−2). We also show that fH (t) = 2 Ω(t h) when H = K h,h , and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function fC 4 (t) for the subgraph version. Motivated by Cygan et al. [Inf. Comput. 2017], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V (H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function fH (t) for every connected graph H on h vertices: if h ≤ 2 the problem can be solved in polynomial time; if h ≥ 3, fH (t) = 2 Θ(t) if H is a clique, and fH (t) = 2 Θ(t h−2) otherwise. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms; Theory of computation → Graph algorithms analysis; Theory of computation → Parameterized complexity and exact algorithm

A biased random-key genetic algorithm for the chordal completion problem

A graph is chordal if all its cycles of length greater than or equal to four contain a chord, i.e., an edge connecting two nonconsecutive vertices of the cycle. Given a graph G = (V, E), the chordal completion problem consists in finding the minimum set of edges to be added to G to obtain a chordal graph. It has applications in sparse linear systems, database management and computer vision programming. In this article, we developed a biased random-key genetic algorithm (BRKGA) for solving the chordal completion problem, based on the strategy of manipulating permutations that represent perfect elimination orderings of triangulations. Computational results show that the proposed heuristic improve the results of the constructive heuristics fill-in and min-degree. We also developed a strategy for injecting externally constructed feasible solutions coded as random keys into the initial population of the BRKGA that significantly improves the solutions obtained and may benefit other implementations of biased random-key genetic algorithms

Reducing Graph Transversals via Edge Contractions

For a graph parameter π, the Contraction(π) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where π is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ℋ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ℋ, which in particular imply that Contraction(π) is co-NP-hard even for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when π is the size of a minimum vertex cover, the problem is in XP parameterized by d