Max-Cut via Kuramoto-type Oscillators

TL;DR

This paper introduces a new functional for the Max-Cut problem inspired by Kuramoto oscillators, providing approximation guarantees and highlighting the NP-hardness of finding near-global minima.

Contribution

It replaces the cosine term with a new function to achieve near-optimal Max-Cut solutions and proves NP-hardness of approximating such minima.

Findings

New functional yields (1-ε)-approximate Max-Cut solutions.

Proves NP-hardness of finding approximate minima of these energy functionals.

Suggests algorithms based on the new functional for practical Max-Cut approximation.

Abstract

We consider the Max-Cut problem. Let $G = (V,E)$ be a graph with adjacency matrix $(a_{ij})_{i,j=1}^{n}$. Burer, Monteiro & Zhang proposed to find, for $n$ angles $\left\{\theta_1, \theta_2, \dots, \theta_n\right\} \subset [0, 2\pi]$, minima of the energy $$ f(\theta_1, \dots, \theta_n) = \sum_{i,j=1}^{n} a_{ij} \cos{(\theta_i - \theta_j)}$$ because configurations achieving a global minimum leads to a partition of size 0.878 Max-Cut(G). This approach is known to be computationally viable and leads to very good results in practice. We prove that by replacing $\cos{(\theta_i - \theta_j)}$ with an explicit function $g_{\varepsilon}(\theta_i - \theta_j)$ global minima of this new functional lead to a $(1-\varepsilon)$Max-Cut(G). This suggests some interesting algorithms that perform well. It also shows that the problem of finding approximate global minima of energy functionals of this type is NP-hard in general.