Weighted Random Popular Matchings

For a set A of n applicants and a set I of m items, let us consider the problem of matching applicants to items, where each applicant x ∈ A provides its preference list defined on items. We say that an applicant x prefers an item p than an item q if p is located at higher position than q in its preference list. For any matchingsM andM′ of the matching problem, we say that an applicant x prefersM overM′ if x prefersM(x) overM′(x). For the matching problem, we say thatM is more popular thanM′ if the number of applicants preferringM overM′ is larger than the number of applicants preferringM′ overM, and defineM to be a popular matching if there are no other matchings that are more popular thanM. Assume that A is partitioned into A1, A2, . . . , Ak and each Ai is assigned a weight wi such that w1 > w2 > · · · > wk > 0. For such a matching problem, we say that M is more popular thanM′ if the total weight of applicants preferringM overM′ is larger than the total weight of applicants preferringM′ overM, and defineM to be a k-weighted popular matching if there are no other matchings that are more popular thanM. Mahdian showed that if m > 1.42n, then a random instance of the matching problem has a popular matching with high probability, but nothing is known for the k-weighted matching problems. In this paper, we analyze the k-weighted matching problems, and we show that for any β such that m = βn, (lower bound) if β/n = o(1), then a random instance of the 2-weighted matching problems does not have a 2-weighted popular matching with probability 1− o(1); (upper bound) if n/β = o(1), then a random instance of the 2-weighted matching problems has a 2-weighted popular matching with probability 1− o(1).