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.

翻译：我们考虑Max-$3$-Section问题，给定一个无向图$G=(V,E)$，其边赋有非负权重$w :E\rightarrow \mathbb{R}_+$，目标是将顶点集$V$划分为三个大小相等的部分，并最大化不同部分之间边的总权重。Max-$3$-Section与其他被广泛研究的图划分问题（例如Max-$k$-Cut、Max-$3$-Cut和Max-Bisection）密切相关。我们提出一个多项式时间算法，其近似比为$0.795$，改进了此前最优的$0.673$近似比。多个部分具有相等大小的要求使得Max-$3$-Section比Max-Bisection等问题更难处理。我们展示了一种新算法，该算法结合了Lasserre层次结构的现有方法以及一种随机割策略，足以得出我们的结果。