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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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References

  1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 434-443. IEEE, 2014. Google Scholar
  2. Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic parameterized problems and algorithms. ACM Trans. Algorithms, 16(4), July 2020. URL: https://doi.org/10.1145/3395037.
  3. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '21, pages 522-539, USA, 2021. Society for Industrial and Applied Mathematics. URL: https://dl.acm.org/doi/10.5555/3458064.3458096.
  4. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, July 1995. URL: https://doi.org/10.1145/210332.210337.
  5. Max Bannach, Zacharias Heinrich, Rüdiger Reischuk, and Till Tantau. Dynamic Kernels for Hitting Sets and Set Packing. In Petr A. Golovach and Meirav Zehavi, editors, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), volume 214 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:18, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.7.
  6. Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), pages 23:1-23:18, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.23.
  7. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: Fast subset convolution. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 67-74, New York, NY, USA, 2007. Association for Computing Machinery. URL: https://doi.org/10.1145/1250790.1250801.
  8. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. Journal of Computer and System Sciences, 87:119-139, 2017. URL: https://doi.org/10.1016/j.jcss.2017.03.003.
  9. Cornelius Brand. Patching Colors with Tensors. In 27th Annual European Symposium on Algorithms (ESA 2019), volume 144 of Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1-25:16, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.25.
  10. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 151-164, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3188745.3188902.
  11. Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartłomiej Wróblewski, et al. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 796-809. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.50.
  12. Rajesh Chitnis, Graham Cormode, Mohammad Taghi Hajiaghayi, and Morteza Monemizadeh. Parameterized streaming: Maximal matching and vertex cover. In Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1234-1251, 2015. URL: https://doi.org/10.1137/1.9781611973730.82.
  13. Richard A. Demillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193-195, 1978. URL: https://doi.org/10.1016/0020-0190(78)90067-4.
  14. Zdeněk Dvořák, Martin Kupec, and Vojtěch Tůma. A dynamic data structure for mso properties in graphs with bounded tree-depth. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014, pages 334-345, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-662-44777-2_28.
  15. Zdeněk Dvořák and Vojtěch Tůma. A dynamic data structure for counting subgraphs in sparse graphs. In Proceedings of the 13th International Conference on Algorithms and Data Structures, WADS'13, pages 304-315, Berlin, Heidelberg, 2013. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-40104-6_27.
  16. Kathrin Hanauer, Monika Henzinger, and Christian Schulz. Recent advances in fully dynamic graph algorithms. CoRR, abs/2102.11169, 2021. URL: http://arxiv.org/abs/2102.11169.
  17. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 21-30, 2015. Google Scholar
  18. Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional hardness for sensitivity problems. CoRR, 2017. URL: http://arxiv.org/abs/1703.01638.
  19. Yoichi Iwata and Keigo Oka. Fast dynamic graph algorithms for parameterized problems. ArXiv, abs/1404.7307, 2014. URL: http://arxiv.org/abs/1404.7307.
  20. Ioannis Koutis. Faster algebraic algorithms for path and packing problems. In Automata, Languages and Programming, pages 575-586, Berlin, Heidelberg, 2008. URL: https://doi.org/10.1007/978-3-540-70575-8_47.
  21. Ioannis Koutis. Constrained multilinear detection for faster functional motif discovery. Information Processing Letters, 112(22):889-892, November 2012. URL: https://doi.org/10.1016/j.ipl.2012.08.008.
  22. Ioannis Koutis and Ryan Williams. Limits and applications of group algebras for parameterized problems. ACM Trans. Algorithms, 12(3), May 2016. URL: https://doi.org/10.1145/2885499.
  23. Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 603-610, 2010. Google Scholar
  24. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, October 1980. URL: https://doi.org/10.1145/322217.322225.
  25. Dekel Tsur. Faster deterministic parameterized algorithm for k-path. Theoretical Computer Science, 790:96-104, 2019. URL: https://doi.org/10.1016/j.tcs.2019.04.024.
  26. Ryan Williams. Finding paths of length k in 𝒪^*(2^k) time. Inf. Process. Lett., 109(6):315-318, February 2009. URL: https://doi.org/10.1016/j.ipl.2008.11.004.
  27. Michał Włodarczyk. Clifford algebras meet tree decompositions. Algorithmica, 81(2):497-518, February 2019. URL: https://doi.org/10.1007/s00453-018-0489-3.
  28. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM '79, pages 216-226. Springer-Verlag, 1979. URL: https://doi.org/10.5555/646670.698972.
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