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.ICALP.2022.9
URN: urn:nbn:de:0030-drops-163504
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16350/
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Alman, Josh ; Hirsch, Dean

Parameterized Sensitivity Oracles and Dynamic Algorithms Using Exterior Algebras

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LIPIcs-ICALP-2022-9.pdf (0.8 MB)


Abstract

We design the first efficient sensitivity oracles and dynamic algorithms for a variety of parameterized problems. Our main approach is to modify the algebraic coding technique from static parameterized algorithm design, which had not previously been used in a dynamic context. We particularly build off of the "extensor coding" method of Brand, Dell and Husfeldt [STOC'18], employing properties of the exterior algebra over different fields.
For the k-Path detection problem for directed graphs, it is known that no efficient dynamic algorithm exists (under popular assumptions from fine-grained complexity). We circumvent this by designing an efficient sensitivity oracle, which preprocesses a directed graph on n vertices in 2^k poly(k) n^{Ļ‰+o(1)} time, such that, given š“ updates (mixing edge insertions and deletions, and vertex deletions) to that input graph, it can decide in time š“Ā² 2^kpoly(k) and with high probability, whether the updated graph contains a path of length k. We also give a deterministic sensitivity oracle requiring 4^k poly(k) n^{Ļ‰+o(1)} preprocessing time and š“Ā² 2^{Ļ‰ k + o(k)} query time, and obtain a randomized sensitivity oracle for the task of approximately counting the number of k-paths. For k-Path detection in undirected graphs, we obtain a randomized sensitivity oracle with O(1.66^k nĀ³) preprocessing time and O(š“Ā³ 1.66^k) query time, and a better bound for undirected bipartite graphs.
In addition, we present the first fully dynamic algorithms for a variety of problems: k-Partial Cover, m-Set k-Packing, t-Dominating Set, d-Dimensional k-Matching, and Exact k-Partial Cover. For example, for k-Partial Cover we show a randomized dynamic algorithm with 2^k poly(k)polylog(n) update time, and a deterministic dynamic algorithm with 4^k poly(k)polylog(n) update time. Finally, we show how our techniques can be adapted to deal with natural variants on these problems where additional constraints are imposed on the solutions.

BibTeX - Entry

@InProceedings{alman_et_al:LIPIcs.ICALP.2022.9,
  author =	{Alman, Josh and Hirsch, Dean},
  title =	{{Parameterized Sensitivity Oracles and Dynamic Algorithms Using Exterior Algebras}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16350},
  URN =		{urn:nbn:de:0030-drops-163504},
  doi =		{10.4230/LIPIcs.ICALP.2022.9},
  annote =	{Keywords: sensitivity oracles, k-path, dynamic algorithms, parameterized algorithms, set packing, partial cover, exterior algebra, extensor, algebraic algorithms}
}

Keywords: sensitivity oracles, k-path, dynamic algorithms, parameterized algorithms, set packing, partial cover, exterior algebra, extensor, algebraic algorithms
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022


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