Phase transitions in theq-coloring of random hypergraphs

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 ≥ 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.

Biased landscapes for random constraint satisfaction problems

“…The most important additional feature unveiled by these phase diagrams is that for some values of the parameters k, α, , there exits (at least) two different non-trivial solutions of the 1RSB equations at m = 1 (38). This type of behavior was described in [70] for a family of random CSPs generalizing the hypergraph bicoloring, and its consequences for inference problems (or planted CSPs) have been discussed in [69]. In order to reach numerically these different solutions we used the population dynamics algorithm explained in Sec.…”

“…In particular for k = 4 the model has a continuous phase transition and the SA algorithm seems to be very efficient in this case: the algorithmic threshold α alg (k = 4) ≈ 4.7 is well beyond the dynamic threshold α d,u = 4.083 and not far from the 1RSB estimate of the satisfiability threshold α sat (k = 4) ≈ 4.9 [70] (this is only expected to be an upperbound on the true satisfiability threshold due to an instability towards higher levels of RSB). On the contrary for k ≥ 5 the phase transition taking place at α d,u is of the random first order type (discontinuous) and this seems to The fast decays of u0 as a function of τ for α = αopt and = opt confirms that for these "optimal" parameters SA is effective in finding the ground state in linear time.…”

“…The first non-trivial moment is thus the variance, which is found after a short computation (see for instance [69] or App. B in [70] for more details) to grow if and only if αk(k − 1)θ 2 > 1, where θ is the derivative of g(u 1 , . .…”

“…have a dramatic effect on the performance of SA, which is able to find solutions only slightly beyond α d,u , stopping far from the α sat threshold. For example, for k = 5 we have α d,u = 9.465, α alg ≈ 9.6 and α sat = 10.46 [70].…”

Rigid Colorings of Hypergraphs and Contiguity

“…In particular, at a critical threshold known as the clustering threshold, the solution space shatters into exponentially many, exponentially small clusters that are well separated. As we increase the density further, it is predicted [16] that we see the emergence of frozen variables which take the same value for every solution within the cluster. We show that at a critical density known as the rigidity threshold, a typical solution possesses a linear number of frozen variables.…”