Finding Optimal Triangulations Parameterized by Edge Clique Cover

Peer reviewe

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Publisher Copyright: © 2022, The Author(s).We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (cc) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem cc is at most the number of taxa, in fractional hypertreewidth cc is at most the number of hyperedges, and in treewidth of Bayesian networks cc is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2 cc, the number of potential maximal cliques is at most 3 cc, and these objects can be listed in times O∗(2 cc) and O∗(3 cc) , respectively, even when no edge clique cover is given as input; the O∗(·) notation omits factors polynomial in the input size. These enumeration algorithms imply O∗(3 cc) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O∗(4 m) time and O∗(3 m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size cc′ is given as a part of the input we give O∗(2cc′) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O∗(2 n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O∗(9cc′) and O∗(9cc+O(log2cc)) for problems in this framework.Peer reviewe

TREEWIDTH and PATHWIDTH parameterized by vertex cover

After the number of vertices, Vertex Cover is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k) time. This complements recent polynomial kernel results for TREEWIDTH and PATHWIDTH parameterized by the Vertex Cover

Hardness of Approximation for H-Free Edge Modification Problems: Towards a Dichotomy

For a fixed graph H, the H-free Edge Deletion/Completion/Editing problem asks to delete/add/edit the minimum possible number of edges in G to get a graph that does not contain an induced subgraph isomorphic to the graph H. In this work, we investigate H-free Edge Deletion/Completion/Editing problems in terms of the hardness of their approximation. We formulate a conjecture according to which all the graphs with at least five vertices can be classified into several groups of graphs with specific structural properties depending on the hardness of approximation for the corresponding H-free Edge Deletion/Completion/Editing problems. Also, we make significant progress in proving that conjecture by showing that it is sufficient to resolve it only for a finite set of graphs

Revisiting Interval Graphs for Network Science

The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear structure in the system. The generalization is an embedding of a graph onto a multi-dimensional Euclidean space and it was used by scientists to study the multi-relational complexity of ecology. However the research went out of fashion in the 1980s and was not revisited when Network Science recently expressed interests with multi-relational networks known as multiplexes. This paper studies interval graphs from the perspective of Network Science

On the (non-)existence of polynomial kernels for Pl-free edge modification problems

Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E \vartriangle F) satisfies the property P. In the P edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free and Cl-free edge-deletion problems for large enough l