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In this paper we study the solution space structure of model RB, a standardprototype of Constraint Satisfaction Problem (CSPs) with growing domains. Usingrigorous the first and the second moment method, we show that in the solvablephase close to the satisfiability transition, solutions are clustered intoexponential number of well-separated clusters, with each cluster containssub-exponential number of solutions. As a consequence, the system has aclustering (dynamical) transition but no condensation transition. This pictureof phase diagram is different from other classic random CSPs with fixed domainsize, such as random K-Satisfiability (K-SAT) and graph coloring problems,where condensation transition exists and is distinct from satisfiabilitytransition. Our result verifies the non-rigorous results obtained using cavitymethod from spin glass theory, and sheds light on the structures of solutionspaces of problems with a large number of states.

Solution space structure of random constraint satisfaction problems with growing domains