An algorithm for fractional assignment problems

Abstract

In this paper, we propose a polynomial-time algorithm for fractional assignment problems. The fractional assignment problem is interpreted as follows. Let G = (I, J, E) be a bipartite graph where I and J are vertex sets and E ⊂- I × J is an edge set. We call an edge subset X(⊂- E) an assignment if every vertex is incident to exactly one edge from X. Given an integer weight cij and a positive integer weight dij for every edge (i, j)ε{lunate} E, the fractional assignment problem finds an assignment X(⊂- E) such that the ratio (∑(i, j)ε{lunate}Xcij) (∑(i, j)ε{lunate}Xdij) is minimized. Some algorithms were developed for the fractional assignment problem. Recently, Radzik (1992) showed that an algorithm which is based on the parametric approach and Newton's method is the fastest one among them. Actually, it solves the fractional assignment problem in O( nmlog2(nCD) (1 + log log(nCD) - log log(nD))) time where |I|=|J|=n, |E|=m, C = max{1, max{|cij|:(i, j)ε{lunate} E}} and D = max{dij: (i, J)ε{lunate} E} + 1. Our algorithm developed in this paper is also based on the parametric approach, but it is combined with the approximate binary search method. The time complexity of our algorithm is O( nm log D log(nCD)) time, and provides with a better time bound than the above algorithm. © 1995.