Solution space structure of random constraint satisfaction problems with growing domains

TL;DR

This paper rigorously analyzes the solution space of the model RB constraint satisfaction problem with growing domains, revealing a clustering transition without condensation, differing from fixed-domain CSPs.

Contribution

It provides the first rigorous proof of the clustering structure in model RB with growing domains, confirming non-rigorous spin glass predictions.

Findings

Solutions form exponential clusters near the satisfiability threshold

No condensation transition occurs in the solution space

Results differ from fixed-domain CSPs like K-SAT and graph coloring

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 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.