Learning Variable Ordering Heuristics for Solving Constraint Satisfaction Problems

Abstract--Backtracking search algorithms are often used to solve the Constraint Satisfaction Problem (CSP). The efficiency of backtracking search depends greatly on the variable ordering heuristics. Currently, the most commonly used heuristics are handcrafted 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. We show that directly optimizing the search cost is hard for bootstrapping, and propose to optimize the expected cost of reaching a leaf node in the search tree. T o 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 the learned policies outperform classical handcrafted heuristics in terms of minimizing the search tree size, and can effectively generalize to instances that are larger than those used in training. Constraint Satisfaction Problem (CSP) is one of the most widely studied problems in computer science and artificial intelligence. It provides a common framework for modeling and solving combinatorial problems in many application domains, such as planning and scheduling [1], [2], vehicle routing [3], [4], graph problems [5], [6], and computational biology [7], [8]. A CSP instance involves a set of variables and constraints. T o solve it, one needs to find a value assignment for all variables such that all constraints are satisfied, or prove such assignment does not exist. Despite its ubiquitous applications, unfortunately, CSP is well known to be NPcomplete in general [9]. T o solve CSP efficiently, backtracking search algorithms are often employed, which are exact algorithms with the guarantee that a solution will be found if one exists.