Exploiting Low-Rank Structure in Max-K-Cut Problems

TL;DR

This paper introduces a novel, scalable algorithm for Max-3-Cut leveraging low-rank matrix structures, outperforming classical SDP relaxations in efficiency while maintaining competitive results.

Contribution

It develops a low-rank based algorithm for Max-3-Cut that guarantees inclusion of the optimal solution and offers theoretical approximation bounds.

Findings

Algorithm achieves performance comparable to existing methods.

Approach is highly scalable and parallelizable.

Guarantees include exact solution inclusion for low-rank cases.

Abstract

We approach the Max-3-Cut problem through the lens of maximizing complex-valued quadratic forms and demonstrate that low-rank structure in the objective matrix can be exploited, leading to alternative algorithms to classical semidefinite programming (SDP) relaxations and heuristic techniques. We propose an algorithm for maximizing these quadratic forms over a domain of size $K$ that enumerates and evaluates a set of $O\left(n^{2r-1}\right)$ candidate solutions, where $n$ is the dimension of the matrix and $r$ represents the rank of an approximation of the objective. We prove that this candidate set is guaranteed to include the exact maximizer when $K=3$ (corresponding to Max-3-Cut) and the objective is low-rank, and provide approximation guarantees when the objective is a perturbation of a low-rank matrix. This construction results in a family of novel, inherently parallelizable and theoretically-motivated algorithms for Max-3-Cut. Extensive experimental results demonstrate that our approach achieves performance comparable to existing algorithms across a wide range of graphs, while being highly scalable.