On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT

TL;DR

This paper explores the complexity of the Hitting Set problem under various structural graph parameters, providing a fixed-parameter tractable algorithm for paths in graphs with bounded cyclomatic number using 2-SAT, and establishing hardness results for other cases.

Contribution

It introduces a fixed-parameter algorithm for hitting paths in graphs with bounded cyclomatic number leveraging 2-SAT, and analyzes the problem's complexity under different structural parameters.

Findings

Algorithm for hitting paths in graphs with cyclomatic number k

Connection of the problem to 2-SAT in multiple valued logic

Hardness results for other structural parameterizations

Abstract

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 2^{5k} (|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.