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.

翻译：本文通过最大化复值二次型的方法研究最大3割问题，并证明目标矩阵的低秩结构可被有效利用，从而提出替代经典半定规划松弛和启发式技术的算法。我们提出一种在规模为$K$的定义域上最大化此类二次型的算法，该算法通过枚举并评估$O\left(n^{2r-1}\right)$个候选解实现，其中$n$为矩阵维度，$r$表示目标函数近似矩阵的秩。我们证明当$K=3$（对应最大3割问题）且目标函数为低秩时，该候选解集必然包含精确最优解；当目标函数为低秩矩阵的扰动时，我们给出了近似保证。这一构造产生了一系列新颖、具有内在并行性且理论驱动的最大3割问题算法。大量实验结果表明，我们的方法在各类图结构上均能达到与现有算法相当的性能，同时具备高度可扩展性。