Deriving Information from Sampling and Diving

Abstract

We investigate the impact of sampling and diving in the solution of constraint satisfaction problems. A sample is a complete assignment of variables to values taken from their domain according to a a given distribution. Diving consists in repeatedly performing depth first search attempts with random variable and value selection, constraint propagation enabled and backtracking disabled; each attempt is called a dive and, unless a feasible solution is found, it is a partial assignment of variables (whereas a sample is a –possibly infeasible– complete assignment). While the probability of finding a feasible solution via sampling or diving is negligible if the problem is difficult enough, samples and dives are very fast to generate and, intuitively, even when they are infeasible, they give some statistic information on search space structure. The aim of this paper is to understand to what extent it is possible to help the CSP solving process with information derived from sampling and diving. In particular, we are interested in extracting from samples and dives precise indications on how good/bad are individual variable-value assignments with respect to feasibility. We formally prove that even uniform sampling could provide precise evaluation of the quality of variable-value assignments; as expected, this requires huge sample sizes and is therefore not useful in practice. On the contrary, diving seems to be much better suited for assignment evaluation purposes. Three dive features are identified and evaluated on a collection of Partial Latin Square instances, showing that diving provides information that can be fruitfully exploited. Many promising direction for future research are proposed.