Rishikesh Gajjala

We give an almost complete characterization of the hardness of c-coloring χ-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that: beginenumerate ıtem We give a new distributed algorithm that finds a c-coloring in χ-chromatic graphs in tildemathcalO(n^frac1alpha) rounds, with alpha = biglłfloorfracc-1chi - 1bigrrfloor. ıtem We prove that any distributed algorithm for this problem requires Ømega(n^frac1alpha) rounds. endenumerate Our upper bound holds in the classical, deterministic mathsfLOCAL model, while the near-matching lower bound holds in the emphnon-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, captures all models of distributed graph algorithms that obey physical causality; this includes not only classical deterministic mathsfLOCAL and randomized mathsfLOCAL but also mathsfquantum-mathsfLOCAL, even with a pre-shared quantum state. We also show that similar arguments can be used to prove that, e.g., 3-coloring 2-dimensional grids or c-coloring trees remain hard problems even for the non-signaling model, and in particular do not admit any quantum advantage. Our lower-bound arguments are purely graph-theoretic at heart; no background on quantum information theory is needed to establish the proofs.