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

2026-02-23

AI Summary

This work proposes a novel approach to the Max-3-Cut problem based on complex-valued quadratic optimization, leveraging the low-rank structure of the objective matrix to circumvent conventional semidefinite programming relaxations and heuristic strategies. By enumerating and evaluating $O(n^{2r-1})$ candidate solutions—where $r$ denotes the approximate rank of the objective matrix—the authors establish, for the first time, that the global optimum is guaranteed to reside within this candidate set when $K=3$ and the objective matrix is low-rank. Theoretical guarantees are also provided for approximately low-rank cases. The algorithm is inherently parallelizable and achieves performance comparable to state-of-the-art methods across various graph structures while demonstrating superior scalability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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.

Problem

• Max-3-Cut
• low-rank structure
• quadratic optimization
• combinatorial optimization

Innovation

• low-rank structure
• Max-3-Cut
• complex-valued quadratic forms
• combinatorial optimization
• parallelizable algorithms