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Parameterized Sensitivity Oracles and Dynamic Algorithms Using Exterior Algebras

Authors Josh Alman, Dean Hirsch

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Author Details

Josh Alman
  • Department of Computer Science, Columbia University, New York, NY, USA
Dean Hirsch
  • Department of Computer Science, Columbia University, New York, NY, USA

Funding

This research was supported in part by a grant from the Simons Foundation (Grant Number 825870 JA), by NSF grant CCF-2107187, by a grant from the Columbia-IBM center for Blockchain and Data Transparency, by JPMorgan Chase & Co., and by LexisNexis Risk Solutions. Any views or opinions expressed herein are solely those of the authors listed.

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Josh Alman and Dean Hirsch. Parameterized Sensitivity Oracles and Dynamic Algorithms Using Exterior Algebras. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 9:1-9:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.9

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 𝓁 updates (mixing edge insertions and deletions, and vertex deletions) to that input graph, it can decide in time 𝓁² 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 𝓁² 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(𝓁³ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • sensitivity oracles
  • k-path
  • dynamic algorithms
  • parameterized algorithms
  • set packing
  • partial cover
  • exterior algebra
  • extensor
  • algebraic algorithms

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