A Fixed-Parameter Algorithm for the Schrijver Problem

04/19/2022

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by   Ishay Haviv, et al.

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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.