Counting Solutions of Constraint Satisfiability Problems:Exact Phase Transitions and Approximate Algorithm

TL;DR

This paper investigates the phase transition phenomenon in counting solutions of #CSP problems, precisely locating critical points and proposing an approximate algorithm for solution estimation.

Contribution

It establishes the existence and exact location of phase transitions in #CSP and introduces an approximate algorithm for solution count estimation.

Findings

Phase transition in #CSP exists as variables grow large.

Critical points align with theoretical predictions.

Proposed algorithm effectively estimates solution counts.

Abstract

The study of phase transition phenomenon of NP complete problems plays an important role in understanding the nature of hard problems. In this paper, we follow this line of research by considering the problem of counting solutions of Constraint Satisfaction Problems (#CSP). We consider the random model, i.e. RB model. We prove that phase transition of #CSP does exist as the number of variables approaches infinity and the critical values where phase transitions occur are precisely located. Preliminary experimental results also show that the critical point coincides with the theoretical derivation. Moreover, we propose an approximate algorithm to estimate the expectation value of the solutions number of a given CSP instance of RB model.