Near-optimal approximation algorithm for simultaneous Max-Cut

TL;DR

This paper presents a polynomial-time approximation algorithm for the simultaneous Max-Cut problem that achieves a near-optimal approximation ratio of 0.8780, improving significantly over previous results.

Contribution

It introduces a novel approach using a stronger Sum-of-Squares SDP relaxation and advanced rounding techniques to surpass the natural SDP's integrality gap.

Findings

Achieves a 0.8780 approximation factor for simultaneous Max-Cut.

Uses Sum-of-Squares hierarchy SDP relaxation and Raghavendra-Tan rounding.

Improves upon the previous 1/2 - o(1) approximation bound.

Abstract

In the simultaneous Max-Cut problem, we are given $k$ weighted graphs on the same set of $n$ vertices, and the goal is to find a cut of the vertex set so that the minimum, over the $k$ graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of $1/2 - o(1)$ for this problem (and an approximation factor of $1/2 + \epsilon_k$ in the unweighted case, where $\epsilon_k \rightarrow 0$ as $k \rightarrow \infty$). In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation factor of $0.8780$ (for all constant $k$). The natural SDP formulation for simultaneous Max-Cut was shown to have an integrality gap of $1/2+\epsilon_k$ in [BKS15]. In achieving the better approximation guarantee, we use a stronger Sum-of-Squares hierarchy SDP relaxation and a rounding algorithm based on Raghavendra-Tan [RT12], in addition to techniques from [BKS15].