License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2021.7
URN: urn:nbn:de:0030-drops-153900
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Bannach, Max ; Heinrich, Zacharias ; Reischuk, Rüdiger ; Tantau, Till

Dynamic Kernels for Hitting Sets and Set Packing

LIPIcs-IPEC-2021-7.pdf (0.8 MB)


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

  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},
  address =	{Dagstuhl, Germany},
  URL =		{},
  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

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