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 = 1 case) is equivalent to finding a homomorphism to the Kneser graph KGa,b. It is tightly connected with the fractional chromatic number, and has multiple applications within computer science. We study the complexity of determining whether a graph has an (a:b)-coloring. As shown by Cygan et al. [SODA 2016], given an arbitrary n-vertex graph G and h-vertex graph H one cannot determine in time 2 whether G admits a homomorphism to H , unless the Exponential Time Hypothesis (ETH) fails. Despite the fact that when H is the Kneser graph KGa,b we have h = ( a b ) , Nederlof [2008] showed a (b + 1) · poly(n)-time algorithm for (a:b)coloring. Our main result is that this is essentially optimal: there is no algorithm with running time 2 b)·n unless the ETH fails. The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [Canad. Math. Bull., 1965], 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 also prove that the running time of the algorithms of Abasi et al. [MFCS 2014] and of Gabizon et al. [ESA 2015] for the r-monomial detection problem are optimal under ETH. Work supported by the National Science Centre of Poland, grants number 2013/11/D/ST6/03073 (MP, MW) and 2015/17/N/ST6/01224 (AS). The work of Ł. Kowalik is a part of the project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677651). Michał Pilipczuk is supported by the Foundation for Polish Science (FNP) via the START stipend programme. CNRS, LaBRI, France University of Warsaw, Poland