 
Summary: A 0.5APPROXIMATION 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 dicutmax dicut with given sizes of parts or max
dicut with gspwhose 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 0:5approximation 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 satises 0 < Æ < 1=2 and is the same for all components.
Key words. approximation algorithm, directed cut, linear relaxation, basic solution
AMS subject classications. 68W25, 05C85, 90C27, 90C35
1. Introduction. Let G be a directed graph. A directed cut in G is dened
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
