New Applications of Clique Separator Decomposition for the Maximum Weight Stable Set Problem
نویسندگان
چکیده
Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem; our results improve and extend various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are P5-free (the P5 is the induced path with five vertices) and obtain new polynomial time results for the MWS problem. The complexity of this problem on the class of P5-free graphs is still unknown. c © 2006 Elsevier B.V. All rights reserved.
منابع مشابه
Algorithmic Aspects of Switch Cographs
This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n) decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomp...
متن کاملDecomposition techniques applied to the Clique-Stable set separation problem
In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K ∩S = ∅, there exists a cut (W,W ′) in C such that K ⊆W and S ⊆W ′. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [20] asked in 1...
متن کاملOn Clique Separators, Nearly Chordal Graphs, and the Maximum Weight Stable Set Problem
Clique separators in graphs are a helpful tool used by Tarjan as a divideand-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Maximum Clique, Graph Coloring and Minimum Fill-in but few examples of graph classes having clique separators are known. We use this method to solve MWS in polynomial time for two classes where the unweighted Maximu...
متن کاملMaximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull
The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a ch...
متن کاملThe Multi-terminal Vertex Separator Problem: Polytope Characterization and TDI-ness
In this paper we discuss a variant of the well-known k-separator problem. Consider the simple graph G = (V ∪T,E) with V ∪T the set of vertices, where T is a set of distinguished vertices called terminals, inducing a stable set and E a set of edges. Given a weight function w : V → N, the multi-terminal vertex separator problem consists in finding a subset S ⊆ V of minimum weight intersecting eve...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005