NeuroPrim: An Attention-based Model for Solving NP-hard Spanning Tree Problems

TL;DR

NeuroPrim is an attention-based neural framework that effectively solves complex spanning tree problems on graphs by leveraging a Markov Decision Process and Prim's algorithm, outperforming heuristics and generalizing well to large instances.

Contribution

The paper introduces NeuroPrim, a novel neural approach that formulates spanning tree problems as an MDP and applies Prim's algorithm to reduce complexity, enabling effective learning and generalization.

Findings

Outperforms strong heuristics on literature instances.

Achieves small optimality gaps up to 250 vertices.

Demonstrates strong generalization to instances as large as 1000 vertices.

Abstract

Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios, often requiring intricate algorithmic design and exponential time. Recently, there has been growing interest in end-to-end deep neural networks for solving routing problems. However, such methods typically produce sequences of vertices, which makes it difficult to apply them to general combinatorial optimization problems where the solution set consists of edges, as in various spanning tree problems. In this paper, we propose NeuroPrim, a novel framework for solving various spanning tree problems by defining a Markov Decision Process (MDP) for general combinatorial optimization problems on graphs. Our approach reduces the action and state space using Prim's algorithm and trains the resulting model using REINFORCE. We apply our framework to three difficult problems on Euclidean space: the Degree-constrained Minimum Spanning Tree (DCMST) problem, the Minimum Routing Cost Spanning Tree (MRCST) problem, and the Steiner Tree Problem in graphs (STP). Experimental results on literature instances demonstrate that our model outperforms strong heuristics and achieves small optimality gaps of up to 250 vertices. Additionally, we find that our model has strong generalization ability, with no significant degradation observed on problem instances as large as 1000. Our results suggest that our framework can be effective for solving a wide range of combinatorial optimization problems beyond spanning tree problems.