Tight lower bounds for the complexity of multicoloring

TL;DR

This paper establishes tight lower bounds on the computational complexity of the multicoloring problem, showing it cannot be solved faster than certain exponential bounds unless ETH fails, and introduces novel combinatorial tools for such proofs.

Contribution

The paper proves new tight lower bounds for multicoloring complexity using detecting matrices, refining the understanding of graph homomorphism problems under ETH.

Findings

Multicoloring problem cannot be solved in time $f(b)\cdot 2^{o(\log b)\cdot n}$ unless ETH fails.

Graph homomorphism problem does not admit a $2^{O(n+h)}$ algorithm unless ETH fails.

Algorithms for r-monomial detection are optimal under ETH.

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 running time $f(b)\cdot 2^{o(\log b)\cdot n}$, for any computable $f(b)$, unless the Exponential Time Hypothesis (ETH) fails. A $(b+1)^n\cdot \text{poly}(n)$-time algorithm due to Nederlof [2008] 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 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. [SODA 2016]. The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindstr\"om [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 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.