An Analysis of a Decomposition Heuristic for the Assignment Problem

J.M. Kurtzherg proposed a method of obtaining approximate solutions to the assignment problem by decomposing a large problem into many smaller subproblems. Thus a km × km assignment problem is decomposed into k 2 problems of size m × m and one prt~blem of size k × k. In this paper we analyze the performance of this heuristic, obtaining the following main results: It) For the maximization problem, the ratio of the optimal solution to the heuristic solution can be as lalg¢ as. bat cannot e~teeed rain( k, m); 4 2) For the minimization problem, it k = o~ ~1/log n I where n ~ nd,', and the matrix elements are independently drawn from a uniform distribution on [0, t), in the limit t he expected value of the heuristic solution is at least k/3 times that of the optimal solution.