The Probabilistic Analysis of a Heuristic for the Assignment Problem

We present a heuristic to solve the $m \times m$ assignment problem in $O(m^2 )$ time. The assignment problem is formulated as a weighted complete bipartite graph $G = (S,T,E)$, $|S| = |T| = m$. For convenience we assume that m is even. The main procedure in the heuristic is to construct a graph $G_d = (S,T,E_d )$ which is a subgraph of G, $|E_d | = 4dn$, $n = {m / 2}$, such that we can find a perfect matching in $G_d $ with probability at least $1 - \frac{1}{3}({d / n})^{d^2 - 4d - 1} $. An $O(|S||E|)$ exact algorithm is used to find a minimum weight matching M in $G_d $. Any unmatched vertices in G relative to M are then matched by a greedy algorithm. The expected value of the total cost of the matching found by the heuristic is shown to be less than six if the costs are independent and identically distributed uniformly in the unit interval. Further, with the above probability, the heuristic produces a solution which is at most six times the optimal solution.