When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2021.7
URN: urn:nbn:de:0030-drops-153900
URL: https://drops.dagstuhl.de/opus/volltexte/2021/15390/
 Go to the corresponding LIPIcs Volume Portal

Dynamic Kernels for Hitting Sets and Set Packing

 pdf-format:

Abstract

Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a size-k hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size k^d (which is a polynomial kernel size as d is a constant), and they do so in time m⋅ 2^d poly(d) for a small polynomial poly(d) (which is a linear runtime as d is again a constant).
We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-k^d hitting set kernel in memory (including moments when no size-k hitting set exists). This paper presents a deterministic solution with worst-case time 3^d poly(d) for updating the kernel upon hyperedge inserts and time 5^d poly(d) for updates upon deletions. These bounds nearly match the time 2^d poly(d) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a dynamic hitting set kernel with constant, deterministic, worst-case update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k hitting sets in d-hypergraphs with update times O(1) and query times O(c^k) where c = d - 1 + O(1/d) equals the best base known for the static setting.

BibTeX - Entry

```@InProceedings{bannach_et_al:LIPIcs.IPEC.2021.7,
author =	{Bannach, Max and Heinrich, Zacharias and Reischuk, R\"{u}diger and Tantau, Till},
title =	{{Dynamic Kernels for Hitting Sets and Set Packing}},
booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
pages =	{7:1--7:18},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-216-7},
ISSN =	{1868-8969},
year =	{2021},
volume =	{214},
editor =	{Golovach, Petr A. and Zehavi, Meirav},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/15390},
URN =		{urn:nbn:de:0030-drops-153900},
doi =		{10.4230/LIPIcs.IPEC.2021.7},
annote =	{Keywords: Kernelization, Dynamic Algorithms, Hitting Set, Set Packings}
}```

 Keywords: Kernelization, Dynamic Algorithms, Hitting Set, Set Packings Collection: 16th International Symposium on Parameterized and Exact Computation (IPEC 2021) Issue Date: 2021 Date of publication: 22.11.2021

DROPS-Home | Fulltext Search | Imprint | Privacy