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Kernelizing the Hitting Set Problem in Linear Sequential and Constant Parallel Time

Authors Max Bannach, Malte Skambath, Till Tantau

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Author Details

Max Bannach
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany
Malte Skambath
  • Department of Computer Science, Kiel University, Germany
Till Tantau
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany

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Max Bannach, Malte Skambath, and Till Tantau. Kernelizing the Hitting Set Problem in Linear Sequential and Constant Parallel Time. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SWAT.2020.9

Abstract

We analyze a reduction rule for computing kernels for the hitting set problem: In a hypergraph, the link of a set c of vertices consists of all edges that are supersets of c. We call such a set critical if its link has certain easy-to-check size properties. The rule states that the link of a critical c can be replaced by c. It is known that a simple linear-time algorithm for computing hitting set kernels (number of edges) at most k^d (k is the hitting set size, d is the maximum edge size) can be derived from this rule. We parallelize this algorithm and obtain the first AC⁰ kernel algorithm that outputs polynomial-size kernels. Previously, such algorithms were not even known for artificial problems. An interesting application of our methods lies in traditional, non-parameterized approximation theory: Our results imply that uniform AC⁰-circuits can compute a hitting set whose size is polynomial in the size of an optimal hitting set.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Kernelization
  • Approximation
  • Hitting Set
  • Constant-Depth Circuits

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