Generalizations Of The Popular Matching Problem
Abstract
Matching problems arise in several realworld scenarios like assigning posts to applicants, houses to trainees and roommates to one another. In this thesis we consider the bipartite matching problem where one side of the bipartition specifies preferences over the other side. That is, we
are given a bipartite graph G = (A ∪ P,E) where A denotes the set of applicants, P denotes the set of posts, and the preferences of applicants are specified by ranks on the edges. Several notions of optimality like paretooptimality, rankmaximality, popularity have been studied in the literature; we focus on the notion of popularity. A matching M is more popular than another matching M′ if the number of applicants that prefer M to M′ exceeds the number of applicants that prefer M′ to M. A matching M is said to be popular if there exists no matching that is more popular than M. Popular matchings have the desirable property that no applicant majority can force a migration to another matching. However, popular matchings do not provide a complete answer since there exist simple instances that do not admit any
popular matching. Abraham et al. (SICOMP 2007) characterized instances that admit a
popular matching and also gave efficient algorithms to find one when it exists. We present several generalizations of the popular matchings problem in this thesis.
Majority of our work deals with instances that do not admit any popular matching. We
propose three different solution concepts for such instances. A reasonable solution when an instance does not admit a popular matching is to output a matching that is least unpopular amongst the set of unpopular matchings. McCutchen (LATIN 2008) introduced and studied measures of unpopularity, namely the unpopularity factor and unpopularity margin. He proved that computing either a least unpopularity factor matching or a least unpopularity margin matching is NPhard. We build upon this work and design an O(km√n) time algorithm which produces matchings with bounded unpopularity provided a certain subgraph of G admits an Acomplete matching (a matching that matches all the applicants). Here n and m denote the number of vertices and the number of edges in G respectively, and k, which is bounded by A,
is the number of iterations taken by our algorithm to terminate. We also show that if a certain subgraph of G admits an Acomplete matching, then we have computed a matching with the least unpopularity factor.
Another feasible solution for instances without any popular matching is to output a mixed matching that is popular. A mixed matching is simply a probability distribution over the set of matchings. A mixed matching Q is popular if no mixed matching is more popular than Q. We
seek to answer the existence and computation of popular mixed matchings in a given instance G. We begin with a linear programming formulation to compute a mixed matching with the least unpopularity margin. We show that although the linear program has exponentially many constraints, we have a polynomial time separation oracle and hence a least unpopularity margin mixed matching can be computed in polynomial time. By casting the popular mixed matchings problem as a two player zerosum game, it is possible to prove that every instance of the popular matchings problem admits a popular mixed matching. Therefore, the matching returned by our linear program is indeed a popular mixed matching.
Finally, we propose augmentation of the input graph for instances that do not admit any popular matching. Assume that we are dealing with a set of items B (say, DVDs/books) instead of posts and it is possible to make duplicates of these items. Our goal is to make duplicates of appropriate items such that the augmented graph admits a popular matching. However, since allowing arbitrarily many copies for items is not feasible in practice, we impose restrictions in two forms – (i) associating costs with items, and (ii) bounding the number of copies. In the first case, we assume that we pay a price of cost(b) for every extra copy of b that we make; the first
copy is assumed to be given to us at free. The total cost of the augmented instance is the sum of costs of all the extra copies that we make. Our goal is to compute a minimum cost augmented instance which admits a popular matching. In the second case, along with the input graph G = (A ∪ B,E), we are given a vector hc1, c2, . . . , cBi denoting upper bounds on the number of copies of every item. We seek to answer whether there exists a vector hx1, x2, . . . , xBi such that having xi copies of item bi where 1 ≤ xi ≤ ci enables the augmented graph to admit a popular matching. We prove that several problems under both these models turn out to be NPhard – in fact they remain NPhard even under severe restrictions on the preference lists.
Our final results deal with instances that admit popular matchings. When the input instance admits a popular matching, there may be several popular matchings – in fact there may be several maximum cardinality popular matchings. Hence one may not be content with returning any maximum cardinality popular matching and instead ask for an optimal popular matching. Assuming that the notion of optimality is specified as a part of the problem, we present an
O(m + n21 ) time algorithm for computing an optimal popular matching in G. Here m denotes
the number of edges in G and n1 denotes the number of applicants. We also consider the
problem of computing a minimum cost popular matching where with every post p, a price
cost(p) and a capacity cap(p) are associated. A post with capacity cap(p) can be matched with up to cap(p) many applicants. We present an O(mn1) time algorithm to compute a minimum cost popular matching in such instances.
We believe that the work provides interesting insights into the popular matchings problem
and its variants.
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