Online Matchings via Randomized Queueing

We study a very simple and natural algorithm for finding a matching of “hats” with “men”: line up the men in a random order, then, taking the hats in random order, assign each hat to the first man still in line who wants it, removing the man and hat. The competitive ratio of this algorithm, defined as the expected size of the matching obtained, divided by the size of the maximum matching possible, is not known. We prove that, in the case of n men and n hats, where hat i is acceptable to man j whenever i ≤ j, the competitive ratio is nearly optimal, specifically 1−O(n−1/5). This input has been a “hard case” for a number of other online matching algorithms. We conjecture that, in general, this algorithm has a competitive ratio of 1 − o(1) for any acceptability graph whose minimum degree tends to infinity. This may be the simplest known algorithm for “random arrival order online matching” for which such a conjecture has been made. We also conjecture that the worst-case competitive ratio of our algorithm is 1 − 1/e, which is also optimal. ∗Department of Computer Science, University of New Mexico, Albuquerque, NM 87108, U.S.A. Email: