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

08/07/2023

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by   Dor Katzelnick, et al.

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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→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.