Random Almost-Popular Matchings
نویسندگان
چکیده
For a set A of n people and a set B ofm items, with each person having a preference list that ranks all items from most wanted to least wanted, we consider the problem of matching every person with a unique item. A matchingM is called -popular if for any other matching M ′, the number of people who prefer M ′ to M is at most n plus the number of those who prefer M to M ′. In 2006, Mahdian showed that when randomly generating people’s preference lists, if m/n > 1.42, then a 0-popular matching exists with 1 − o(1) probability; and if m/n < 1.42, then a 0-popular matching exists with o(1) probability. The ratio 1.42 can be viewed as a transition point, at which the probability rises from asymptotically zero to asymptotically one, for the case = 0. In this paper, we introduce an upper bound and a lower bound of the transition point in more general cases. In particular, we show that when randomly generating each person’s preference list, if α(1 − e−1/α) > 1 − , then an -popular matching exists with 1− o(1) probability (upper bound); and if α(1− e−(1+e 1/α)/α) < 1− 2 , then an -popular matching exists with o(1) probability (lower bound).
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