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

TL;DR

This work proposes an unsupervised learning framework combined with heuristics for MMCP that can provide valid and high-quality solutions and demonstrates the superiority of the method against two techniques designed.

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.

Figures (13)

• Figure 1: An example to illustrate the difference among max-cut, maximum minimal cut, and connected maximum cut.
• Figure 2: Overview of the PIONEER pipeline. (a) Remove all bridges to induce connected subgraphs. (b) Utilize the unsupervised solver with a performance guarantee to obtain discrete solutions. (c) Construct spanning trees based on the discrete solutions. (d) Further enhance the solutions through the heuristic solver. (e) Compare all solutions to determine the optimal solution.
• Figure 3: The pipeline of unsupervised solver. The parameter $\theta$ is learned by graph neural networks, trained through graph instances without bridges. For testing, the network outputs relaxed solutions that are rounded to discrete solutions.
• Figure 4: An instance to show the synthetic datasets. The vertices in the graph correspond to random pictures of Mnist, whose values $v_i$ are the one-digit numbers that match the pictures. And the weight of edge $e_{ij}$ is equal to $v_i + v_j +v_i \cdot v_j$.
• Figure 5: Effectiveness of the unsupervised solver. We report the results on the synthetic dataset here. The bars denote the cut values, and the lines denote the execution time.

Theorems & Definitions (15)

• Definition 1: The Maximum Minimal Cut Problem
• Theorem 1
• Theorem 2: Eigenvalue Interlacing for Laplacian
• Definition 2: Deterministic Rounding
• Theorem 3: Performance Guarantee
• Theorem 4
• Definition 3: Disconnected-Vertex
• proof
• Theorem 5: Weyl's inequality
• Corollary 1