License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2020.9
URN: urn:nbn:de:0030-drops-122566
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12256/
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### Kernelizing the Hitting Set Problem in Linear Sequential and Constant Parallel Time

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### 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.

### BibTeX - Entry

```@InProceedings{bannach_et_al:LIPIcs:2020:12256,
author =	{Max Bannach and Malte Skambath and Till Tantau},
title =	{{Kernelizing the Hitting Set Problem in Linear Sequential and Constant Parallel Time}},
booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
pages =	{9:1--9:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-150-4},
ISSN =	{1868-8969},
year =	{2020},
volume =	{162},
editor =	{Susanne Albers},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12256},
URN =		{urn:nbn:de:0030-drops-122566},
doi =		{10.4230/LIPIcs.SWAT.2020.9},
annote =	{Keywords: Kernelization, Approximation, Hitting Set, Constant-Depth Circuits}
}
```

 Keywords: Kernelization, Approximation, Hitting Set, Constant-Depth Circuits Collection: 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020) Issue Date: 2020 Date of publication: 12.06.2020

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