Learning Variable Ordering Heuristics for Solving Constraint Satisfaction Problems

TL;DR

This paper introduces a deep reinforcement learning approach using graph neural networks to automatically learn variable ordering heuristics for CSPs, outperforming traditional heuristics and generalizing to larger instances.

Contribution

It presents a novel method combining deep RL and GNNs to discover adaptive variable ordering heuristics for CSPs, improving search efficiency.

Findings

Learned heuristics outperform classical ones in reducing search tree size.

The approach generalizes well to larger, unseen CSP instances.

Optimizing expected search cost effectively guides heuristic discovery.

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 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. 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. 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 the learned policies outperform classical hand-crafted heuristics in terms of minimizing the search tree size, and can effectively generalize to instances that are larger than those used in training.