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

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.Comment: Presented at the 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was corrected in this update.

Regular Encodings from Max-CSP into Partial Max-SAT *

Abstract We define a number of origina

A Modular Reduction of Regular Logic to Classical Logic

In this paper we first define a reduction that transforms an instance of Regular-SAT into a satisfiability equivalent instance of SAT. The reduction has interesting properties: (i) the size of is linear in the size of , (ii) transforms regular Horn formulas into Horn formulas, and (iii) transforms regular 2-CNF formulas into 2-CNF formulas. Second, we describe a new satisfiability testing algorithm that determines the satisfiability of a regular 2-CNF formula in time O(j j log j j); this algorithm is inspired by the reduction . Third, we introduce the concept of renamable-Horn regular CNF formula and define another reduction 0 that transforms a renamable-Horn instance of Regular-SAT into a renamable-Horn instance 0 of SAT. We use this reduction to show that both membership and satisfiability of renamable-Horn regular CNF formulas can be decided in time O(j j log j j). 1 Introduction The satisfiability problem of regular CNF formulas, or Regular-SAT, has attracted the inte..