Document

# A Fixed-Parameter Algorithm for the Kneser Problem

## File

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

## Acknowledgements

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

## Cite As

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

## 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.
2. Karol Borsuk. Drei Sätze über die n-dimensionale euklidische Sphäre. Fundamenta Mathematicae, 20(1):177-190, 1933.
3. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
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.
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.
6. Xiaotie Deng, Zhe Feng, and Rucha Kulkarni. Octahedral Tucker is PPA-complete. Electronic Colloquium on Computational Complexity (ECCC), 24:118, 2017.
7. Irit Dinur and Ehud Friedgut. Intersecting families are essentially contained in juntas. Comb. Probab. Comput., 18(1-2):107-122, 2009.
8. Paul Erdös, Chao Ko, and Richard Rado. Intersection theorems for systems of finite sets. Quart. J. Math., 12(1):313-320, 1961.
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.
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.
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.
12. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019.
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.
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.
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.
17. Martin Kneser. Aufgabe 360. Jahresbericht der Deutschen Mathematiker-Vereinigung, 58(2):27, 1955.
18. Andrey Kupavskii. Diversity of uniform intersecting families. Eur. J. Comb., 74:39-47, 2018.
19. László Lovász. Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory, Ser. A, 25(3):319-324, 1978.
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.
21. Jiří Matoušek. A combinatorial proof of Kneser’s conjecture. Combinatorica, 24(1):163-170, 2004.
22. Jiří Matoušek. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Springer Publishing Company, Incorporated, 2007.
23. Jiří Matoušek and Günter M. Ziegler. Topological lower bounds for the chromatic number: A hierarchy. Jahresbericht der DMV, 106(2):71-90, 2004.
24. Colin McDiarmid. Concentration. In Probabilistic methods for algorithmic discrete mathematics, volume 16 of Algorithms Combin., pages 195-248. Springer, Berlin, 1998.
25. Nimrod Megiddo and Christos H. Papadimitriou. On total functions, existence theorems and computational complexity. Theor. Comput. Sci., 81(2):317-324, 1991.
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.
27. Alexander Schrijver. Vertex-critical subgraphs of Kneser graphs. Nieuw Arch. Wiskd., 26(3):454-461, 1978.
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.