Learning variable ordering heuristics for solving Constraint Satisfaction Problems

Abstract

Backtracking search algorithms are often used to solve the Constraint Satisfaction Problem (CSP), which is widely applied in various domains such as automated planning and scheduling. The efficiency of backtracking search depends greatly on the variable ordering heuristics. Currently, the most commonly used heuristics are hand-crafted based on expert knowledge. In this paper, we propose a deep reinforcement learning based approach to automatically discover new variable ordering heuristics that are better adapted for a given class of CSP instances, without the need of relying on hand-crafted features and heuristics. We show that directly optimizing the search tree size is not convenient for learning, and propose to optimize the expected cost of reaching a leaf node in the search tree. To capture the complex relations among the variables and constraints, we design a representation scheme based on Graph Neural Network that can process CSP instances with different sizes and constraint arities. Experimental results on random CSP instances show that on small and medium sized instances, the learned policies outperform classical hand-crafted heuristics with smaller search tree (up to 10.36% reduction). Moreover, without further training, our policies directly generalize to instances of larger sizes and much harder to solve than those in training, with even larger reduction in the search tree size (up to 18.74%).