For a set @math of @math people and a set @math of @math items, with each person having a preference list that ranks some items in order of preference, we consider the problem of matching every person with a unique item. A matching @math is popular if for any other matching @math , the number of people who prefer @math to @math is not less than the number of those who prefer @math to @math . For given @math and @math , consider the probability of existence of a popular matching when each person's preference list is independently and uniformly generated at random. Previously, Mahdian showed that when people's preference lists are strict (containing no ties) and complete (containing all items in @math ), if @math , where @math is the root of equation @math , then a popular matching exists with probability @math ; and if @math , where @math is the root of equation @math , then a popular matching exists with probability @math ; and if @math , then a popular matching exists with probability @math .
We consider the problem of choosing the best matching of people to positions based on preferences expressed by the people, for which many different optimality criteria have been proposed. A matching is popular if no other matching beats it in a majority vote of the people. The popularity criterion has a manipulation-resistance property, but unfortunately, some sets of preferences admit no popular matching. In this paper, we introduce the least-unpopularity-factor and least-unpopularity-margin criteria, two generalizations of popularity that preserve the manipulation-resistance property but give an optimal matching for every set of preferences. Under each of these generalizations, we show that the "badness" of a given matching can be calculated efficiently but it is NP-hard to find an optimal matching.  We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching Mis popular if there is no matching Mi¾? such that more people prefer Mi¾? to Mthan the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied in [2]. If there is no popular matching, a reasonable substitute is a matching whose unpopularityis bounded. We consider two measures of unpopularity - unpopularity factordenoted by u(M) and unpopularity margindenoted by g(M). McCutchen recently showed that computing a matching Mwith the minimum value of u(M) or g(M) is NP-hard, and that if Gdoes not admit a popular matching, then we have u(M) i¾? 2 for all matchings Min G. Here we show that a matching Mthat achieves u(M) = 2 can be computed in @math time (where mis the number of edges in Gand nis the number of nodes) provided a certain graph Hadmits a matching that matches all people. We also describe a sequence of graphs: H= H 2 , H 3 ,...,H k such that if H k admits a matching that matches all people, then we can compute in @math time a matching Msuch that u(M) ≤ ki¾? 1 and @math . Simulation results suggest that our algorithm finds a matching with low unpopularity.