Phase transitions in the $q$-coloring of random hypergraphs

TL;DR

This paper investigates the solution structure and phase transitions in random hypergraph coloring problems, providing explicit predictions and revealing complex behaviors like multiple RSB solutions and thresholds for various parameters.

Contribution

It offers the first detailed analysis of phase transitions in hypergraph coloring, including explicit thresholds and the discovery of multiple 1RSB solutions.

Findings

Colorability thresholds for K=3 and K=4 hypergraph coloring are not given by 1RSB analysis.

Existence of coexistence of two different 1RSB solutions for q=2, K≥4.

Asymptotic expansions for phase transition densities as q and/or K diverge.

Abstract

We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes $K$ variables that must be assigned one out of $q$ colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs ($K=2$) and bicoloring of random hypergraphs ($q=2$), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size $q$ of the domain of the variables and the arity $K$ of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem ($q=2$) where we find that for $K=3$ and $K=4$ the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q=2$, $K \ge 4$. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where $q$ and/or $K$ diverge.