Papers
Topics
Authors
Recent
Search
2000 character limit reached

Fast Deterministic Chromatic Number under the Asymptotic Rank Conjecture

Published 7 Apr 2024 in cs.DS | (2404.04987v2)

Abstract: In this paper we further explore the recently discovered connection by Bj\"{o}rklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an $n$-vertex graph can be computed deterministically in $O(1.99982n)$ time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the $2n\operatorname{poly}(n)$ time algorithm for chromatic number by Bj\"{o}rklund, Husfeldt, and Koivisto [SICOMP 2009]. Viewed in the other direction, if chromatic number indeed requires deterministic algorithms to run in close to $2n$ time, we obtain a sequence of explicit tensors of superlinear rank, falsifying the asymptotic rank conjecture. Our technique is a combination of earlier algorithms for detecting $k$-colorings for small $k$ and enumerating $k$-colorable subgraphs, with an extension and derandomisation of Pratt's tensor-based algorithm for balanced three-way partitioning to the unbalanced case.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (36)
  1. R. Beigel and D. Eppstein. 3-coloring in time O⁢(1.3289n)𝑂superscript1.3289𝑛{O}(1.3289^{n})italic_O ( 1.3289 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ). Journal of Algorithms, 54(2):168–204, 2005.
  2. A. Björklund and T. Husfeldt. Exact algorithms for exact satisfiability and number of perfect matchings. In M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener, editors, Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, volume 4051 of Lecture Notes in Computer Science, pages 548–559. Springer, 2006.
  3. Trimmed moebius inversion and graphs of bounded degree. Theory Comput. Syst., 47(3):637–654, 2010.
  4. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546–563, 2009.
  5. A. Björklund and P. Kaski. The asymptotic rank conjecture and the set cover conjecture are not both true, 2023. arXiv:2310.11926, to appear in STOC 2024.
  6. C. Bron and J. Kerbosch. Finding all cliques of an undirected graph. Commun. ACM, 16(9):575–577, sep 1973.
  7. Algebraic Complexity Theory. Springer Science & Business Media, 2013.
  8. J. M. Byskov. Algorithms for k-colouring and finding maximal independent sets. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA, pages 456–457. ACM/SIAM, 2003.
  9. J. M. Byskov. Enumerating maximal independent sets with applications to graph colouring. Operations Research Letters, 32(6):547–556, 2004.
  10. On the number of maximal bipartite subgraphs of a graph. Journal of Graph Theory, 48(2):127–132, 2005.
  11. Barriers for fast matrix multiplication from irreversibility. Theory Comput., 17:Paper No. 2, 32, 2021.
  12. N. Christofides. An algorithm for the chromatic number of a graph. Comput. J., 14(1):38–39, 1971.
  13. A piecewise approach for the analysis of exact algorithms, 2024. arXiv:2402.10015.
  14. Rank and border rank of Kronecker powers of tensors and Strassen’s laser method. Comput. Complexity, 31(1):Paper No. 1, 40, 2022.
  15. Towards a geometric approach to Strassen’s asymptotic rank conjecture. Collect. Math., 72(1):63–86, 2021.
  16. D. Eppstein. Small maximal independent sets and faster exact graph coloring. J. Graph Algorithms Appl., 7(2):131–140, 2003.
  17. Improved exact algorithms for counting 3- and 4-colorings. In International Computing and Combinatorics Conference, 2007.
  18. P. A. Gartenberg. Fast Rectangular Matrix Multiplication. PhD thesis, University of California, Los Angeles, 1985.
  19. S. Jukna. Extremal Combinatorics. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg, second edition, 2011. With applications in computer science.
  20. M. Karppa and P. Kaski. Probabilistic tensors and opportunistic boolean matrix multiplication. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 496–515. SIAM, 2019.
  21. J. M. Landsberg. Tensors: Geometry and Applications, volume 128 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.
  22. E. L. Lawler. A note on the complexity of the chromatic number problem. Inf. Process. Lett., 5(3):66–67, 1976.
  23. L. Meijer. 3-coloring in time O⁢(1.3217n)𝑂superscript1.3217𝑛{O}(1.3217^{n})italic_O ( 1.3217 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ), 2023. arXiv:2302.13644.
  24. M. Mitzenmacher and E. Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005.
  25. J. Moon and L. Moser. On cliques in graphs. Israel J. Math., 3:23–28, 1965.
  26. K. Pratt. A stronger connection between the asymptotic rank conjecture and the set cover conjecture, 2023. arXiv:2311.02774, to appear in STOC 2024.
  27. I. Schiermeyer. Deciding 3-colourability in less than O⁢(1.415n)𝑂superscript1.415𝑛{O}(1.415^{n})italic_O ( 1.415 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) steps. In J. van Leeuwen, editor, Graph-Theoretic Concepts in Computer Science, 19th International Workshop, WG ’93, Utrecht, The Netherlands, June 16-18, 1993, Proceedings, volume 790 of Lecture Notes in Computer Science, pages 177–188. Springer, 1993.
  28. V. Strassen. The asymptotic spectrum of tensors and the exponent of matrix multiplication. In 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, 27-29 October 1986, pages 49–54. IEEE Computer Society, 1986.
  29. V. Strassen. The asymptotic spectrum of tensors. J. Reine Angew. Math., 384:102–152, 1988.
  30. V. Strassen. Degeneration and complexity of bilinear maps: some asymptotic spectra. J. Reine Angew. Math., 413:127–180, 1991.
  31. V. Strassen. Algebra and complexity. In First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 429–446. Birkhäuser, Basel, 1994.
  32. The worst-case time complexity for generating all maximal cliques and computational experiments. Theoretical Computer Science, 363(1):28–42, 2006. Computing and Combinatorics.
  33. A. Wigderson and J. Zuiddam. Asymptotic spectra: Theory, applications and extensions. Manuscript dated October 24, 2023; available at https://staff.fnwi.uva.nl/j.zuiddam/papers/convexity.pdf, 2023.
  34. F. Yates. The Design and Analysis of Factorial Experiments. Imperial Bureau of Soil Science, 1937.
  35. O. Zamir. Breaking the 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT barrier for 5-coloring and 6-coloring. In N. Bansal, E. Merelli, and J. 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 113:1–113:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.
  36. O. Zamir. Algorithmic applications of hypergraph and partition containers. In B. Saha and R. A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 985–998. ACM, 2023.
Citations (2)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up