Exact Phase Transitions and Approximate Algorithm of #CSP

Ping Huang, Minghao Yin, Ke Xu

Abstract: In the past decade there has been a significant interest in the phase transition of NP-complete or NP-hard problems. However, it seems difficult to obtain the location of the exact phase transition point. (Xu and Li 2000) may be one of the few works that can prove the existence of phase transition and identified the phase transition points exactly. By introducing a revision of the standard CSP random model(Model B) in (Gent el. al. 2001), as we called Model RB, we can prove that the critical value of the phase transition point can be quantified. Moreover, Model RB provides a framework for generating asymptotically hard random constraint satisfaction problems and therefore has been widely used as benchmarks to evaluate the asymptotic behavior of CSP algorithms (Xu el. al. 2007). In this literature, we follow this line of research to study phase transition of counting the solutions of CSP instances following Model RB. Specifically, we consider a decision version of #CSP, called #CSP ( d). That is, deciding whether the instance has at least d satisfying assignments. Note that #CSP( d) can be viewed as a generalization of #3SAT( 2), which was studied in (Bailey el. al. 2007). So #CSP( d) is at least PP-hard. The contribution of our work is as following: 1) We prove the existence of phase transition in Model RB for #CSP( d) can be guaranteed, and the threshold point can be precisely located rather than in the form of some loose but hard won bounds, for instance (Bailey el. al. 2007). 2) A careful analysis of phase transition can lead us to develop an approximate algorithm to estimate the solutions number in Model RB. Unlike other approximate algorithms, the accuracy of our algorithm increases with the increase of the problem scale.