# An Approximation Algorithm for MAX DICUT with Given Sizes of Parts

## Abstract

Given a directed graph G and an edge 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*. It is known that by using semidefinite programming rounding techniques max dicut can be well approximated — the best approximation with a factor of 0.859 is due to Feige and Goemans. Unfortunately, no similar approach is known to be applicable to max dicut with gsp. This paper presents an 0.5- approximation algorithm for solving the problem. The algorithm is based on exploiting structural properties of basic solutions to a linear relaxation in combination with the pipage rounding technique developed in some earlier papers by two of the authors.

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