Clustering at the Phase Transition

Andrew J. Parkes

Many problem ensembles exhibit a phase transition that is associated with a large peak in the average cost of solving the problem instances. However, this peak is not necessarily due to a lack of solutions: indeed the average number of solutions is typically exponentially large. Here, we study this situation within the context of the satisfiability transition in Random SSAT. We find that a significant subclass of instances emerges as we cross the phase transition. These instances are characterized by having about 85-95% of their variables occurring in unary prime implicates (UPIs), with their remaining variables being subject to few constraints. In such instances the models are not randomly distributed but all lie in a cluster that is exponentially large, but still admits a simple description. Studying the effect of UPIs on the local search algorithm WSAT shows that these "single-cluster" instances are harder to solve, and we relate their appearance at the phase transition to the peak in search cost.

This page is copyrighted by AAAI. All rights reserved. Your use of this site constitutes acceptance of all of AAAI's terms and conditions and privacy policy.