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
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15390/
Bannach, Max ;
Heinrich, Zacharias ;
Reischuk, Rüdiger ;
Tantau, Till
Dynamic Kernels for Hitting Sets and Set Packing
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},
address = {Dagstuhl, Germany},
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}
}
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Keywords: |
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Kernelization, Dynamic Algorithms, Hitting Set, Set Packings |
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Collection: |
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16th International Symposium on Parameterized and Exact Computation (IPEC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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22.11.2021 |
