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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: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16350/
Alman, Josh ;
Hirsch, Dean
Parameterized Sensitivity Oracles and Dynamic Algorithms Using Exterior Algebras
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^{ω+o(1)} time, such that, given ? updates (mixing edge insertions and deletions, and vertex deletions) to that input graph, it can decide in time ?² 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^{ω+o(1)} preprocessing time and ?² 2^{ω 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 n³) preprocessing time and O(?³ 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}
}
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Keywords: |
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sensitivity oracles, k-path, dynamic algorithms, parameterized algorithms, set packing, partial cover, exterior algebra, extensor, algebraic algorithms |
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Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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28.06.2022 |
