The solution space structure of random constraint satisfaction problems with growing domains

Abstract

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 some non-rigorous results obtained using cavity method from spin glass theory.PACS numbers: 89.75. Fb, 89.20.Ff

Clustering phase of a general constraint satisfaction problem model d-k-CSP

“…[19], Bethe free entropy was studied, and it was suggested that the RS solution should always be stable locally; thus, the condensation transition should be absent in this model. In article [20], we indeed prove that the clustering phase exists and persists until the satisfiable-unsatisfiable transition point, so the condensation phase do not exist. The solution space of model RB is quite different from that of other CSPs such as K-SAT or Coloring, where the condensation transition is observed [6].…”

“…Clustering range Cluster diameter ≤ Distance among cluster-regions ≥ Number of clusters [20] r0 < r < 1 a2n (b2 − a2)n exponential In Table II [20]. We have the following observations.…”

“…Other conditions [20] k(n) = const, d(n) = n α A solution pair is a sequence of two assignment solutions. The (Hamming) distance between two solutions is the number of variables which have different values in the two solutions.…”

Super Solutions of the Model RB

“…It has been shown that random instances of model RB are hard to solve at the threshold both theoretically [6] and experimentally [7], thus benchmarks based on model RB have been widely used in algorithm competitions. In addition, studies on the statistical mechanics of model RB show that the replica symmetry solution is always stable in the satisfiable phase [10,11], and there is no condensation transition in model RB [8].…”

Exact Phase Transitions of Model RB with Slower-Growing Domains

“…Model RB has been proved to have sharp SAT-UNSAT phase transition and exact threshold points, and can generate hard instances in the phase transition region [12,14]. In addition, it was proved that model RB has a clustering transition but no condensation transition [17,16]. Moreover, randomly generated forced RB instances with one hidden solution are proved to have both similar distribution of solutions and hardness property with unforced satisfiable RB instances [15].…”