A Value Ordering Heuristic for Solving Ultra-Weak Solutions in Minimax Weighted CSPs

Minimax Weighted Constraint Satisfaction Problems (formerly called Quantified Weighted CSPs) are a framework for modeling soft constrained problems with adversarial conditions. In this paper, we study the effects of a value ordering heuristic in solving ultra-weak solutions on top of the alpha beta tree search with constraint propagation. The value ordering heuristic is based on minimax heuristics from adversarial search, which selects values for variables according to the semantic of quantifiers by considering the problem as a two-player zero sum game. In practice, implementing the heuristic requires costs approximations, and we devise three heuristic variants: HUnary, HBinary, and HFullBinary to approximate costs. In particular, we observe that combining these heuristic variants with consistency notions can achieve a better efficiency and a further reduction of search space. We perform experiments on three benchmarks to compare the effects on applying these heuristic variants, and confirm the feasibility and efficiency of our proposal.