A 0.5-approximation algorithm for MAX DICUT with given sizes of parts

Given a directed graph G and an arc weight function w : E(G) → ℝ₊, the maximum directed cut problem (MAX DICUT) is that of finding a directed cut δ(X) with maximum total weight. In this paper we consider a version of MAX DICUT - MAX DICUT with given sizes of parts or MAX DICUT WITH GSP - whose instance is that of MAX DICUT plus a positive integer p, and it is required to find a directed cut δ(X) having maximum weight over all cuts δ(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, δ, 1/2, 1 - δ, 1}, where δ is a constant that satisfies 0 < δ < 1/2 and is the same for all components.