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

Abstract

Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set systemi¾?$$\mathcal {F} $$ over a finite universei¾?U and an integeri¾?t; the question is whether there is a set ofi¾?t elements that intersects every set ini¾?$$\mathcal {F} $$. 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 treei¾?T on vertex seti¾?U such that the sets ini¾?$$\mathcal {F} $$ induce connected subtrees ofi¾?T. We consider the case that there is a treelike graph with vertex seti¾?U such that the sets ini¾?$$\mathcal {F} $$ 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 graphi¾?G with cyclomatic numberi¾?k, a collectioni¾?$$\mathcal {P} $$ of simple paths ini¾?G, and an integeri¾?t, determines in timei¾?$$2^{5k} |G| +|\mathcal {P} |^{{\mathcal {O}}1}$$ whether there is a vertex set of sizei¾?t that hits all paths ini¾?$$\mathcal {P} $$. 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.