Tight Lower Bounds for the Complexity of Multicoloring

Abstract

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 =1 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 )ċ 2 o (log b )ċ n for any computable f(b) unless the Exponential Time Hypothesis (ETH) fails. A ( b +1) n ċ 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 Lindström [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.