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

Ria Stevens, Fangshuo Liao, Barbara Su, Jianqiang Li, Anastasios Kyrillidis

TL;DR

It is demonstrated that low-rank structure in the objective matrix can be exploited, leading to alternative algorithms to classical semidefinite programming (SDP) relaxations and heuristic techniques, and results in a family of novel, inherently parallelizable and theoretically-motivated algorithms for Max-3-Cut.

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.

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

TL;DR

It is demonstrated that low-rank structure in the objective matrix can be exploited, leading to alternative algorithms to classical semidefinite programming (SDP) relaxations and heuristic techniques, and results in a family of novel, inherently parallelizable and theoretically-motivated algorithms for Max-3-Cut.

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 that enumerates and evaluates a set of candidate solutions, where is the dimension of the matrix and represents the rank of an approximation of the objective. We prove that this candidate set is guaranteed to include the exact maximizer when (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.
Paper Structure (35 sections, 30 theorems, 116 equations, 4 figures, 2 tables, 3 algorithms)

This paper contains 35 sections, 30 theorems, 116 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Exact Rank-1 Algorithm
  4. Exact Rank-$r$ Algorithm
  5. Approximate Algorithms for Perturbed Low-Rank Graphs
  6. Experiments
  7. Experimental Setup.
  8. Small-Scale Experiments
  9. Large-Scale Experiments
  10. Very Large Graphs.
  11. Conclusion
  12. Acknowledgments
  13. Derivation of Complex Ternary Quadratic Form for Max-3-Cut
  14. Complex Encoding of Partitions.
  15. Reformulation as a Quadratic Optimization.
  16. ...and 20 more sections

Key Result

Theorem 1

Let ${\mathbf{Q}}^\star = \lambda \mathbf{q} \mathbf{q}^\dagger \in \mathbb{C}^{n}$ be a rank-1, positive semi-definite matrix. Algorithm alg:max3cut_rank1 exactly solves the complex discrete quadratic maximization problem eq:quad_max with objective matrix ${\mathbf{Q}}^\star$ in $\mathcal{O}\left(

Figures (4)

  • Figure 1: A toy example of the partition of the hypercube $\mathcal{H}_r$ formed by $2r-1$ hypersurfaces, with $r=2$. Each region, or cell, of the partition corresponds to a different candidate cut.
  • Figure 2: Empirical approximation ratios of the studied algorithms on 5-regular graphs of varying sizes (left) and on Erdős-Rényi graphs with $n=100$ nodes and varying edge probabilities (right).
  • Figure 3:
  • Figure 4:

Theorems & Definitions (74)

  • Theorem 1
  • proof
  • Remark 2: Parallelizability
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 5
  • proof
  • Lemma C.1: Unit Norm Property
  • proof
  • ...and 64 more