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A Fixed-Parameter Algorithm for the Schrijver Problem

Author Ishay Haviv

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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
  • Schrijver graph
  • Kneser graph
  • Fixed-parameter tractability

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Abstract

The Schrijver graph S(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} that do not include two consecutive elements modulo n, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of S(n,k) is n-2k+2 (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of S(n,k) with n-2k+1 colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class PPA. We prove that it can be solved by a randomized algorithm with running time n^O(1) ⋅ k^O(k), hence it is fixed-parameter tractable with respect to the parameter k.

Cite As Get BibTex

Ishay Haviv. A Fixed-Parameter Algorithm for the Schrijver Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.IPEC.2022.16

Author Details

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

Funding

Research supported in part by the Israel Science Foundation (grant No. 1218/20).

Acknowledgements

We thank Gabriel Istrate for clarifications on [Gabriel Istrate et al., 2021] and the anonymous reviewers for their useful suggestions.

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