On a Greedy Heuristic for Complete Matching * Edward M . Reingold

, Finding a minimum weighted complete matching on a set of vertices in which the distances satisfy the triangle inequality is of general interest and of particular importance when drawing graphs on a mechanical plotter. The "greedy" heuristic of repeatedly matching the two closest unmatched points can be implemented in worst-case time O(n log n), a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to n to find the minimum cost matching in a weighted graph. We show that, for an even number n of vertices whose distances satisfy the triangle inequality, the ratio of the cost of the matching produced by this greedy heuristic to the cost of the minimal matching is at most lg 3n 1, lg 0.58496, and there are examples that achieve this bound. We conclude that this greedy heuristic, although desirable because of its simplicity, would be a poor choice for this problem.