A 0.502$\cdot$MaxCut Approximation using Quadratic Programming

TL;DR

This paper introduces a quadratic programming approach for approximating MaxCut, achieving at least 0.502 approximation ratio, offering a new method that performs well empirically compared to existing algorithms.

Contribution

The paper presents a novel quadratic programming formulation for MaxCut that guarantees at least a 0.502 approximation ratio, expanding the toolkit beyond SDP and spectral methods.

Findings

Achieves at least 0.502 MaxCut approximation ratio.

Performs well in practical experiments.

Provides a new theoretical approach to MaxCut approximation.

Abstract

We study the MaxCut problem for graphs $G=(V,E)$. The problem is NP-hard, there are two main approximation algorithms with theoretical guarantees: (1) the Goemans \& Williamson algorithm uses semi-definite programming to provide a 0.878MaxCut approximation (which, if the Unique Games Conjecture is true, is the best that can be done in polynomial time) and (2) Trevisan proposed an algorithm using spectral graph theory from which a 0.614MaxCut approximation can be obtained. We discuss a new approach using a specific quadratic program and prove that its solution can be used to obtain at least a 0.502MaxCut approximation. The algorithm seems to perform well in practice.