On structural parameterizations of hitting set: Hitting paths in graphs using 2-SAT

作     者：Jansen, Bart M. P. 

作者机构：Department of Mathematics and Computer Science Eindhoven University of Technology Netherlands 

出 版 物：《Journal of Graph Algorithms and Applications》 (J. Graph Algorithms and Appl.)

年 卷 期：2017年第21卷第2期

页      面：219-245页

核心收录：

学科分类：07[理学] 070102[理学-计算数学] 0701[理学-数学] 

基　　金：Supported by NWO Veni grant "Frontiers in Parameterized Preprocessing" and NWO Gravitation grant "Networks". An extended abstract of this work appeared in the proceedings of the 41st International Workshop on Graph-Theoretic Concepts in Computer Science 

主　　题：Combinatorial optimization 

摘      要：Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t;the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs;the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 25k(|G| + |P|)O(1)whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results. © 2017, Brown University. All rights reserved.