Estimating Bounds on Expected Plateau Size in MAXSAT Problems

Abstract

Stochastic local search algorithms can now successfully solve MAXSAT problems with thousands of variables or more. A key to this success is how effectively the search can navigate and escape plateau regions. Furthermore, the solubility of a problem depends on the size and exit density of plateaus, especially those closest to the optimal solution. In this paper we model the plateau phenomenon as a percolation process on hypercube graphs. We develop two models for estimating bounds on the size of plateaus and prove that one is a lower bound and the other an upper bound on the expected size of plateaus at a given level. The models’ accuracy is demonstrated on controlled random hypercube landscapes. We apply the models to MAXSAT through analogy to hypercube graphs and by introducing an approach to estimating, through sampling, a key parameter of the models. Using this approach, we assess the accuracy of our bound estimations on uniform random and structured benchmarks. Surprisingly, we find similar trends in accuracy across random and structured problem instances. Less surprisingly, we find a high accuracy on smaller plateaus with systematic divergence as plateaus increase in size.

This research was sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.