A Fixed-Parameter Algorithm for the Kneser Problem

Author Ishay Haviv



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

Acknowledgements

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

Cite AsGet 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

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