Document Open Access Logo

A Fixed-Parameter Algorithm for the Kneser Problem

Author Ishay Haviv

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2022.72.pdf
  • Filesize: 0.72 MB
  • 18 pages

Document Identifiers

Author Details

Ishay Haviv
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv, Israel

Funding

  • Haviv, Ishay: Research supported in part by the Israel Science Foundation (grant No. 1218/20).

Acknowledgements

We are grateful to Andrey Kupavskii for helpful discussions and to the anonymous reviewers for their very useful suggestions.

Cite As Get BibTex

Ishay Haviv. A Fixed-Parameter Algorithm for the Kneser Problem. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 72:1-72:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.72

Abstract

The Kneser graph K(n,k) is defined for integers n and k with n ≥ 2k as the graph whose vertices are all the k-subsets of {1,2,…,n} where two such sets are adjacent if they are disjoint. A classical result of Lovász asserts that the chromatic number of K(n,k) is n-2k+2. In the computational Kneser problem, we are given an oracle access to a coloring of the vertices of K(n,k) with n-2k+1 colors, and the goal is to find a monochromatic edge. We present a randomized algorithm for the Kneser problem with running time n^O(1) ⋅ k^O(k). This shows that the problem is fixed-parameter tractable with respect to the parameter k. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser graphs.
We also study the Agreeable-Set problem of assigning a small subset of a set of m items to a group of 𝓁 agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances that satisfy 𝓁 ≥ m - O({log m}/{log log m}). We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with an extended access to the input coloring.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Kneser graph
  • Fixed-parameter tractability
  • Agreeable Set

Metrics

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

Related Versions

References

  1. Paul Beame, Stephen A. Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The relative complexity of NP search problems. J. Comput. Syst. Sci., 57(1):3-19, 1998. Preliminary version in STOC'95. Google Scholar
  2. Karol Borsuk. Drei Sätze über die n-dimensionale euklidische Sphäre. Fundamenta Mathematicae, 20(1):177-190, 1933. Google Scholar
  3. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  4. Argyrios Deligkas, John Fearnley, Alexandros Hollender, and Themistoklis Melissourgos. Constant inapproximability for PPA. In Proc. of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC'22), 2022. Google Scholar
  5. Argyrios Deligkas, Aris Filos-Ratsikas, and Alexandros Hollender. Two’s company, three’s a crowd: Consensus-halving for a constant number of agents. In Proc. of the 22nd ACM Conference on Economics and Computation (EC'21), pages 347-368, 2021. Google Scholar
  6. Xiaotie Deng, Zhe Feng, and Rucha Kulkarni. Octahedral Tucker is PPA-complete. Electronic Colloquium on Computational Complexity (ECCC), 24:118, 2017. Google Scholar
  7. Irit Dinur and Ehud Friedgut. Intersecting families are essentially contained in juntas. Comb. Probab. Comput., 18(1-2):107-122, 2009. Google Scholar
  8. Paul Erdös, Chao Ko, and Richard Rado. Intersection theorems for systems of finite sets. Quart. J. Math., 12(1):313-320, 1961. Google Scholar
  9. Aris Filos-Ratsikas and Paul W. Goldberg. Consensus halving is PPA-complete. In Proc. of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC'18), pages 51-64, 2018. Google Scholar
  10. Aris Filos-Ratsikas and Paul W. Goldberg. The complexity of splitting necklaces and bisecting ham sandwiches. In Proc. of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC'19), pages 638-649, 2019. Google Scholar
  11. Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. A topological characterization of modulo-p arguments and implications for necklace splitting. In Proc. of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA'21), pages 2615-2634, 2021. Google Scholar
  12. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. Google Scholar
  13. Peter Frankl and Andrey Kupavskii. Maximal degrees in subgraphs of Kneser graphs. arXiv, abs/2004.08718, 2020. URL: http://arxiv.org/abs/2004.08718.
  14. Paul W. Goldberg, Alexandros Hollender, Ayumi Igarashi, Pasin Manurangsi, and Warut Suksompong. Consensus halving for sets of items. In Proc. 16th Web and Internet Economics International Conference (WINE'20), pages 384-397, 2020. Google Scholar
  15. Ishay Haviv. The complexity of finding fair independent sets in cycles. In 12th Innovations in Theoretical Computer Science Conference (ITCS'21), pages 4:1-4:14, 2021. Google Scholar
  16. Anthony J. W. Hilton and Eric Charles Milner. Some intersection theorems for systems of finite sets. Quart. J. Math., 18(1):369-384, 1967. Google Scholar
  17. Martin Kneser. Aufgabe 360. Jahresbericht der Deutschen Mathematiker-Vereinigung, 58(2):27, 1955. Google Scholar
  18. Andrey Kupavskii. Diversity of uniform intersecting families. Eur. J. Comb., 74:39-47, 2018. Google Scholar
  19. László Lovász. Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory, Ser. A, 25(3):319-324, 1978. Google Scholar
  20. Pasin Manurangsi and Warut Suksompong. Computing a small agreeable set of indivisible items. Artif. Intell., 268:96-114, 2019. Preliminary versions in IJCAI'16 and IJCAI'17. Google Scholar
  21. Jiří Matoušek. A combinatorial proof of Kneser’s conjecture. Combinatorica, 24(1):163-170, 2004. Google Scholar
  22. Jiří Matoušek. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Springer Publishing Company, Incorporated, 2007. Google Scholar
  23. Jiří Matoušek and Günter M. Ziegler. Topological lower bounds for the chromatic number: A hierarchy. Jahresbericht der DMV, 106(2):71-90, 2004. Google Scholar
  24. Colin McDiarmid. Concentration. In Probabilistic methods for algorithmic discrete mathematics, volume 16 of Algorithms Combin., pages 195-248. Springer, Berlin, 1998. Google Scholar
  25. Nimrod Megiddo and Christos H. Papadimitriou. On total functions, existence theorems and computational complexity. Theor. Comput. Sci., 81(2):317-324, 1991. Google Scholar
  26. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. Google Scholar
  27. Alexander Schrijver. Vertex-critical subgraphs of Kneser graphs. Nieuw Arch. Wiskd., 26(3):454-461, 1978. Google Scholar
  28. Forest W. Simmons and Francis Edward Su. Consensus-halving via theorems of Borsuk-Ulam and Tucker. Math. Soc. Sci., 45(1):15-25, 2003. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact