Tight Lower Bounds for the Complexity of Multicoloring

In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b = I case) is equivalent to finding a homomorphism to the Kneser graph KG(a, b) and gives relaxations approaching the fractional chromatic number. We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with runtime f (b) . 2(o(log) (b).n) for any computable f (b) unless the Exponential Time Hypothesis (ETH) fails. A (b + 1)(n) . poly(n)-time algorithm due to Nederlof [33] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2(O(n+h)) algorithm unless the ETH fails even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [9]. The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindstrom [28], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the runtime of the algorithms of Abasi et al. [1] and of Gabizon et al. [14] for the r-monomial detection problem are optimal under the ETH.