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.
Highlights
Hitting Set is a classic problem in combinatorial optimization that asks, given a set system F over a finite universe U, and an integer t, whether there is a set of t elements that intersects every set in F
We consider whether Hitting Set can be solved efficiently if there is a graph G that is close to being a tree, such that all S ∈ F induce connected subgraphs of G
We have not touched upon the issue of computing, given a generic hitting set instance consisting of a set system F over a universe U, how complex graphs on vertex set U must be in which every set in F induces a connected subgraph
Summary
Hitting Set is a classic problem in combinatorial optimization that asks, given a set system F over a finite universe U , and an integer t, whether there is a set of t elements that intersects every set in F. This corresponds to Hitting Set instances where there is a graph G on U such that for all sets S in F, there is a simple path in G on vertex set S We prove that this problem is fixed-parameter tractable and can be solved in time 25k(|G|+|P|)O(1), which is the main algorithmic result in this paper. The W[1]-hardness result for hitting 3-leaf subtrees parameterized by cyclomatic number implies that this problem is W[1]-hard using the smaller parameterizations by treewidth or feedback vertex number It rules out fixedparameter tractability for many parameters that measure how treelike the input graph is.
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