Exact Phase Transitions of Model RB with Slower-Growing Domains

TL;DR

This paper demonstrates that the second moment method can precisely determine phase transitions in random RB constraint satisfaction problems with slower-growing domains, improving understanding of satisfiability thresholds.

Contribution

It provides the first exact phase transition results for RB models with slower-growing domains using an enhanced second moment analysis.

Findings

Exact phase transition established for RB with slower-growing domains

Improved lower bounds on satisfiability thresholds

Highlights limitations of current methods for tighter results

Abstract

The second moment method has always been an effective tool to lower bound the satisfiability threshold of many random constraint satisfaction problems. However, the calculation is usually hard to carry out and as a result, only some loose results can be obtained. In this paper, based on a delicate analysis which fully exploit the power of the second moment method, we prove that random RB instances can exhibit exact phase transition under more relaxed conditions, especially slower-growing domain size. These results are the best by using the second moment method, and new tools should be introduced for any better results.