eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-12
9:1
9:16
10.4230/LIPIcs.SWAT.2020.9
article
Kernelizing the Hitting Set Problem in Linear Sequential and Constant Parallel Time
Bannach, Max
1
Skambath, Malte
2
Tantau, Till
1
Institute for Theoretical Computer Science, Universität zu Lübeck, Germany
Department of Computer Science, Kiel University, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol162-swat2020/LIPIcs.SWAT.2020.9/LIPIcs.SWAT.2020.9.pdf
Kernelization
Approximation
Hitting Set
Constant-Depth Circuits