A 0.5-APPROXIMATION ALGORITHM FOR MAX DICUT WITH GIVEN SIZES OF PARTS Summary: A 0.5-APPROXIMATION ALGORITHM FOR MAX DICUT WITH GIVEN SIZES OF PARTS ALEXANDER AGEEV  , REFAEL HASSIN y , AND MAXIM SVIRIDENKO z Abstract. Given a directed graph G and an arc weight function w : E(G) ! R+ , the maximum directed cut problem (max dicut) is that of nding a directed cut Æ(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 nd a directed cut Æ(X) having maximum weight over all cuts Æ(X) with jXj = p. Our main result is an 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 f0; Æ; 1=2; 1 Æ; 1g where Æ is a constant that satis es 0 < Æ < 1=2 and is the same for all components. Key words. approximation algorithm, directed cut, linear relaxation, basic solution AMS subject classi cations. 68W25, 05C85, 90C27, 90C35 1. Introduction. Let G be a directed graph. A directed cut in G is de ned to be the set of arcs leaving some vertex subset X (we denote it by Æ(X)). Given a directed graph G and an arc weight function w : E(G) ! R+ , the maximum directed cut problem (max dicut) is that of nding a directed cut Æ(X) with maximum total weight. In this paper we consider a version of max dicut (max dicut with given Collections: Mathematics