An Improved Approximation Algorithm for the Max-$3$-Section Problem

TL;DR

This paper introduces a new polynomial-time algorithm for the Max-3-Section problem, achieving a 0.795 approximation ratio, significantly improving upon the previous best of 0.673, by combining Lasserre hierarchy and random cut strategies.

Contribution

The paper presents a novel approximation algorithm for Max-3-Section that surpasses prior bounds by integrating Lasserre hierarchy techniques with a random cut approach.

Findings

Achieves a 0.795 approximation ratio for Max-3-Section.

Improves the previous best approximation of 0.673.

Combines Lasserre hierarchy with random cut strategy effectively.

Abstract

We consider the Max-$3$-Section problem, where we are given an undirected graph $ G=(V,E)$ equipped with non-negative edge weights $w :E\rightarrow \mathbb{R}_+$ and the goal is to find a partition of $V$ into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-$3$-Section is closely related to other well-studied graph partitioning problems, e.g., Max-$k$-Cut, Max-$3$-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of $ 0.795$, that improves upon the previous best known approximation of $ 0.673$. The requirement of multiple parts that have equal sizes renders Max-$3$-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result.