Abstract
Given a hypergraph H = (V,E), what is the smallest subset $X \subseteq V$ such that e ∩ X≠∅ holds for all e ∈ E? This problem, known as the hitting set problem, is a basic problem in combinatorial optimization and has been studied extensively in both classical and parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph H and a number k as input and output a hypergraph H′ such that (1) H has a hitting set of size k if and only if $H^{\prime }$ has such a hitting set and (2) the size of $H^{\prime }$ depends only on k and on the maximum cardinality d of hyperedges in H. The algorithms run in polynomial time and can be parallelized to a certain degree: one can easily compute hitting set kernels in parallel time O(k) and not-so-easily in time O(d) – but it was conjectured that these are the best parallel algorithms possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application.
Highlights
IntroductionIn the present paper we refute the conjecture of Chen et al and show that there is a constant parallel time kernelization algorithm for the hitting set problem: Problem 1.1
The hitting set problem is the following combinatorial problem: Given a hypergraph H = (V, E) as input, consisting of a set V of vertices and a set E of hyperedges with e ⊆ V for all e ∈ E, find a set X ⊆ V of minimum size that “hits” all hyperedges e ∈ E, that is, e ∩ X = ∅
In the present paper we refute the conjecture of Chen et al and show that there is a constant parallel time kernelization algorithm for the hitting set problem: Problem 1.1. pk,d-hitting-set Instance: A hypergraph H = (V, E) and a number k ∈ N
Summary
In the present paper we refute the conjecture of Chen et al and show that there is a constant parallel time kernelization algorithm for the hitting set problem: Problem 1.1. The theorem and corollary imply that all problems that can be reduced to pk,d-hitting-set via a parameter-preserving AC0-reduction admit a kernelization computable by an AC0-circuit family This includes pk-vertex-cover, which is just pk,d-hitting-set with d fixed at 2; pktriangle-removal, where the objective is to remove at most k vertices from an undirected graph so that no triangles remain; and pk,deg-dominating-set, where we must find a dominating set of size at most k in an undirected graph and we parameterized by k and the maximum degree of the vertices. Full proofs can be found in the full version of the paper [5]
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