An Unsupervised Learning Framework Combined with Heuristics for the Maximum Minimal Cut Problem

TL;DR

This paper introduces a novel unsupervised learning framework combined with heuristics to effectively solve the NP-hard Maximum Minimal Cut Problem, leveraging graph neural networks and tree-based heuristics for high-quality solutions.

Contribution

It is the first work to integrate machine learning with heuristics specifically for MMCP, using a relaxation-plus-rounding approach with graph neural networks and tree transformations.

Findings

Outperforms existing techniques in solution quality.

End-to-end trainable unsupervised model.

Efficient heuristic improvements reduce computation time.

Abstract

The Maximum Minimal Cut Problem (MMCP), a NP-hard combinatorial optimization (CO) problem, has not received much attention due to the demanding and challenging bi-connectivity constraint. Moreover, as a CO problem, it is also a daunting task for machine learning, especially without labeled instances. To deal with these problems, this work proposes an unsupervised learning framework combined with heuristics for MMCP that can provide valid and high-quality solutions. As far as we know, this is the first work that explores machine learning and heuristics to solve MMCP. The unsupervised solver is inspired by a relaxation-plus-rounding approach, the relaxed solution is parameterized by graph neural networks, and the cost and penalty of MMCP are explicitly written out, which can train the model end-to-end. A crucial observation is that each solution corresponds to at least one spanning tree. Based on this finding, a heuristic solver that implements tree transformations by adding vertices is utilized to repair and improve the solution quality of the unsupervised solver. Alternatively, the graph is simplified while guaranteeing solution consistency, which reduces the running time. We conduct extensive experiments to evaluate our framework and give a specific application. The results demonstrate the superiority of our method against two techniques designed.