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[2602.20376] Exploiting Low-Rank Structure in Max-K-Cut Problems
Summary
The paper explores the Max-3-Cut problem by leveraging low-rank structures in complex-valued quadratic forms, proposing new algorithms that outperform classical methods while ensuring scalability and parallelizability.
Why It Matters
This research provides innovative algorithms for the Max-3-Cut problem, which is significant in optimization and graph theory. By exploiting low-rank structures, it enhances computational efficiency and offers new insights into algorithm design, which could benefit various applications in data science and machine learning.
Key Takeaways
- Introduces new algorithms for Max-3-Cut leveraging low-rank structures.
- Demonstrates that the proposed methods are scalable and parallelizable.
- Provides performance guarantees and approximation bounds for low-rank objectives.
- Experimental results show comparable performance to existing algorithms.
- Enhances understanding of complex-valued quadratic forms in optimization.
Computer Science > Data Structures and Algorithms arXiv:2602.20376 (cs) [Submitted on 23 Feb 2026] Title:Exploiting Low-Rank Structure in Max-K-Cut Problems Authors:Ria Stevens, Fangshuo Liao, Barbara Su, Jianqiang Li, Anastasios Kyrillidis View a PDF of the paper titled Exploiting Low-Rank Structure in Max-K-Cut Problems, by Ria Stevens and 4 other authors View PDF HTML (experimental) 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 ...
