An Algorithm for Fractional Assignment Problems

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 c ij and a positive integer weight d ij for every edge (i; j) 2 E; the fractional assignment problem nds an assignment X( E) such that the ratio ( P (i;j)2X c ij )=( P (i;j)2X d ij ) is minimized. There developed some algorithms for the fractional assignment problem. Recently, T. Radzik 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( p nm log(nCD) log(nCD) 1+log log(nCD) log log(nD) ) time where jIj = jJ j = n; jEj = m; C = maxf1;maxfjc ij j : (i; j) 2 Egg and D = maxfd ij : (i; j) 2 Eg+ 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( p nm logD log(nCD)) time, and provides with a better time bound than the above algorithm.