An Upper Bound on the Expected Cost of an Optimal Assignment

Abstract

This chapter presents an upper bound on the expected cost of an optimal assignment. An instance of the n × n assignment problem (AP) is specified by an n × n matrix (Cij) of real numbers. When n is fixed and the cij are drawn independently from the uniform distribution over [0,1];, A* becomes a random variable. Call the matrix (Cij) regular if no two distinct subsets of its elements have the same sum. This implies in particular that the optimal assignment σ is unique. Under the stated assumptions about the probability distribution of the cij, the matrix (Cij) is regular with probability 1. The chapter highlights the regular instances of the AP.