Fixed-Parameter Algorithms for the Kneser and Schrijver Problems

SIAM Journal on Computing, Volume 53, Issue 2, Page 287-314, April 2024.
Abstract. The Kneser graph [math] is defined for integers [math] and [math] with [math] as the graph whose vertices are all the [math]-subsets of [math] where two such sets are adjacent if they are disjoint. The Schrijver graph [math] is defined as the subgraph of [math] induced by the collection of all [math]-subsets of [math] that do not include two consecutive elements modulo [math]. It is known that the chromatic number of both [math] and [math] is [math]. In the computational Kneser and Schrijver problems, we are given access to a coloring with [math] colors of the vertices of [math] and [math], respectively, and the goal is to find a monochromatic edge. We prove that the problems admit randomized algorithms with running time [math], hence they are fixed-parameter tractable with respect to the parameter [math]. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser and Schrijver graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of [math] items to a group of [math] 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 with [math]. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with extended access to the input coloring.