Document Open Access Logo

New Lower Bounds and Upper Bounds for Listing Avoidable Vertices

Authors Mingyang Deng, Virginia Vassilevska Williams, Ziqian Zhong



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2022.41.pdf
  • Filesize: 0.74 MB
  • 14 pages

Document Identifiers

Author Details

Mingyang Deng
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Virginia Vassilevska Williams
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Ziqian Zhong
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Cite AsGet BibTex

Mingyang Deng, Virginia Vassilevska Williams, and Ziqian Zhong. New Lower Bounds and Upper Bounds for Listing Avoidable Vertices. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 41:1-41:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.41

Abstract

We consider the problem of listing all avoidable vertices in a given n vertex graph. A vertex is avoidable if every pair of its neighbors is connected by a path whose internal vertices are not neighbors of the vertex or the vertex itself. Recently, Papadopolous and Zisis showed that one can list all avoidable vertices in O(n^{ω+1}) time, where ω < 2.373 is the square matrix multiplication exponent, and conjectured that a faster algorithm is not possible. In this paper we show that under the 3-OV Hypothesis, and thus the Strong Exponential Time Hypothesis, n^{3-o(1)} time is needed to list all avoidable vertices, and thus the current best algorithm is conditionally optimal if ω = 2. We then show that if ω > 2, one can obtain an improved algorithm that for the current value of ω runs in O(n^3.32) time. We also show that our conditional lower bound is actually higher and supercubic, under a natural High Dimensional 3-OV hypothesis, implying that for our current knowledge of rectangular matrix multiplication, the avoidable vertex listing problem likely requires Ω(n^3.25) time. We obtain further algorithmic improvements for sparse graphs and bounded degree graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Avoidable Vertex
  • Fine-Grained Complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443. IEEE Computer Society, 2014. Google Scholar
  2. Ramesh C. Agarwal and Charles S. Burrus. Fast convolution using Fermat number transforms with applications to digital filtering. IEEE Trans. Acoust. Speech Signal Process., ASSP-(2):87-97, 1974. Google Scholar
  3. Donald Aingworth, Chandra Chekuri, Piotr Indyk, and Rajeev Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput., 28(4):1167-1181, 1999. Google Scholar
  4. Josh Alman. Limits on the universal method for matrix multiplication. Theory Comput., 17:1-30, 2021. Google Scholar
  5. Josh Alman and Virginia Vassilevska Williams. Limits on all known (and some unknown) approaches to matrix multiplication. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 580-591. IEEE Computer Society, 2018. Google Scholar
  6. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 522-539. SIAM, 2021. Google Scholar
  7. Andris Ambainis, Yuval Filmus, and François Le Gall. Fast matrix multiplication: Limitations of the coppersmith-winograd method. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 585-593. ACM, 2015. Google Scholar
  8. Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Toward tight approximation bounds for graph diameter and eccentricities. SIAM J. Comput., 50(4):1155-1199, 2021. Google Scholar
  9. Jesse Beisegel, Maria Chudnovsky, Vladimir Gurvich, Martin Milanic, and Mary Servatius. Avoidable vertices and edges in graphs. In Algorithms and Data Structures - 16th International Symposium, WADS 2019, Edmonton, AB, Canada, August 5-7, 2019, Proceedings, volume 11646 of Lecture Notes in Computer Science, pages 126-139. Springer, 2019. Google Scholar
  10. Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, and Chris Umans. On cap sets and the group-theoretic approach to matrix multiplication. Discrete Analysis, 3:27pp., 2017. Google Scholar
  11. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. On the exact complexity of evaluating quantified k-cnf. Algorithmica, 65(4):817-827, 2013. Google Scholar
  12. Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9(3):251-280, 1990. Google Scholar
  13. Mina Dalirrooyfard and Jenny Kaufmann. Approximation algorithms for min-distance problems in dags. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 60:1-60:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  14. Mina Dalirrooyfard, Ray Li, and Virginia Vassilevska Williams. Hardness of approximate diameter: Now for undirected graphs. In Proceedings of FOCS 2021, page to appear, 2021. Google Scholar
  15. Mina Dalirrooyfard, Thuy Duong Vuong, and Virginia Vassilevska Williams. Graph pattern detection: Hardness for all induced patterns and faster noninduced cycles. SIAM J. Comput., 50(5):1627-1662, 2021. Google Scholar
  16. Mina Dalirrooyfard and Nicole Wein. Tight conditional lower bounds for approximating diameter in directed graphs. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1697-1710. ACM, 2021. Google Scholar
  17. G.A. Dirac. On rigid circuit graphs. Abh.Math.Semin.Univ.Hambg., 25:71-76, 1961. Google Scholar
  18. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theor. Comput. Sci., 326(1-3):57-67, 2004. Google Scholar
  19. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  20. Ton Kloks, Dieter Kratsch, and Haiko Müller. Finding and counting small induced subgraphs efficiently. Inf. Process. Lett., 74(3-4):115-121, 2000. Google Scholar
  21. François Le Gall. Powers of tensors and fast matrix multiplication. In ISSAC 2014 - Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pages 296-303. ACM, New York, 2014. Google Scholar
  22. Francois Le Gall and Florent Urrutia. Improved rectangular matrix multiplication using powers of the coppersmith-winograd tensor. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1029-1046. SIAM, 2018. Google Scholar
  23. Ray Li. Settling SETH vs. approximate sparse directed unweighted diameter (up to (NU)NSETH). In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1684-1696. ACM, 2021. Google Scholar
  24. Tatsuo Ohtsuki, Lap Kit Cheung, and Toshio Fujisawa. Minimal triangulation of a graph and optimal pivoting order in a sparse matrix. Journal of Mathematical Analysis and Applications, 54(3):622-633, 1976. Google Scholar
  25. Charis Papadopoulos and Athanasios Zisis. Computing and listing avoidable vertices and paths, 2021. URL: http://arxiv.org/abs/2108.07160.
  26. Donald J. Rose. Triangulated graphs and the elimination process. Journal of Mathematical Analysis and Applications, 32:597-609, 1970. Google Scholar
  27. Jeremy Spinrad. Some open problems. URL: http://dts-web1.it.vanderbilt.edu/~spinrajp//open.html.
  28. Volker Strassen. Gaussian elimination is not optimal. Numer. Math., 13(4):354-356, 1969. Google Scholar
  29. Mikkel Thorup. Near-optimal fully-dynamic graph connectivity. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 343-350. ACM, New York, 2000. Google Scholar
  30. Joris van der Hoeven. Lazy multiplication of formal power series. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC '97, pages 17-20, New York, NY, USA, 1997. Association for Computing Machinery. Google Scholar
  31. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the International Congress of Mathematicians: Rio de Janeiro 2018, pages 3447-3487. World Scientific, 2018. Google Scholar
  32. Virginia Vassilevska Williams and Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. Google Scholar
  33. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. Google Scholar
  34. Raphael Yuster and Uri Zwick. Fast sparse matrix multiplication. ACM Trans. Algorithms, 1(1):2-13, 2005. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail