The condensation transition in random hypergraph 2-coloring

For many random constraint satisfaction problems such as random satisfiability or random graph or hypergraph coloring, the best current estimates of the threshold for the existence of solutions are based on the first and the second moment method. However, in most cases these techniques do not yield matching upper and lower bounds. Sophisticated but non-rigorous arguments from statistical mechanics have ascribed this discrepancy to the existence of a phase transition called condensation that occurs shortly before the actual threshold for the existence of solutions and that affects the combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time that a condensation transition exists in a natural random CSP, namely in random hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment method breaks down strictly \emph{before} the condensation transition. Our proof also yields slightly improved bounds on the threshold for random hypergraph 2-colorability. We expect that our techniques can be extended to other, related problems such as random k-SAT or random graph k-coloring

On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs

Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p

Solution space structure of random constraint satisfaction problems with growing domains

In this paper we study the solution space structure of model RB, a standard prototype of Constraint Satisfaction Problem (CSPs) with growing domains. Using rigorous the first and the second moment method, we show that in the solvable phase close to the satisfiability transition, solutions are clustered into exponential number of well-separated clusters, with each cluster contains sub-exponential number of solutions. As a consequence, the system has a clustering (dynamical) transition but no condensation transition. This picture of phase diagram is different from other classic random CSPs with fixed domain size, such as random K-Satisfiability (K-SAT) and graph coloring problems, where condensation transition exists and is distinct from satisfiability transition. Our result verifies the non-rigorous results obtained using cavity method from spin glass theory, and sheds light on the structures of solution spaces of problems with a large number of states.Comment: 8 pages, 1 figure

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which type of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem, e.g. dynamical, rigidity and SAT-UNSAT transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance we show that a smoothed version of a decimation strategy based on Belief Propagation is able to find solutions belonging to sub-dominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen. These non-equilibrium solutions belong to the most probable unfrozen clusters.Comment: 38 pages, 10 figure