Constructive Bounds and Exact Expectations for the Random Assignment Problem

The random assignment problem is to choose a minimum-cost perfect matching in a complete n x n bipartite graph, whose edge weights are chosen randomly from some distribution such as the exponential distribution with mean 1. In this case it is known that the expectation does not grow unboundedly with n, but approaches some limiting value c ? between 1.51 and 2. The limit is conjectured to be ß 2 =6, while a recent conjecture has it that for finite n, the expected cost is P n i=1 1=i 2 . This paper contains two principal results. First, by defining and analyzing a constructive algorithm, we show that the limiting expectation is c ? ! 1:94. Second, we extend the finite-n conjecture to partial assignments on complete m x n bipartite graphs, and prove it in some limited cases