A 0.5-Approximation Algorithm for MAX DICUT with Given Sizes of Parts

Abstract

Given a directed graph G and an arc weight function $w: E(G)\rightarrow\mathbb{R}_+$, the maximum directed cut problem ({\sc max dicut}) is that of finding a directed cut $\delta (X)$ with maximum total weight. In this paper we consider a version of {\sc max dicut}---{\sc max dicut} with given sizes of parts or {\sc max dicut with gsp}---whose instance is that of {\sc max dicut} plus a positive integer p, and it is required to find a directed cut $\delta (X)$ having maximum weight over all cuts $\delta (X)$ with $|X|=p$. Our main result is a $0.5$-approximation algorithm for solving the problem. The algorithm is based on a tricky application of the pipage rounding technique developed in some earlier papers by two of the authors and a remarkable structural property of basic solutions to a linear relaxation. The property is that each component of any basic solution is an element of a set $\{0,\delta,1/2,1-\delta,1 \}$, where $\delta$ is a constant that satisfies $0 < \delta < 1/2$ and is the same for all components.