Rank Maximal Matchings -- Structure and Algorithms

Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rank-maximal matchings is #P-Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rank-maximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem

The stable roommates problem with globally-ranked pairs

We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, they can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of an algorithm due to [Irving et al. 06] to a nonbipartite setting. Also, we describe several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs

Efficient algorithms for weighted rank-maximal matchings and related problems

We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph $G =({\mathcal{A}} \cup {\mathcal{P}}, {\mathcal{E}})$ , with a partition of the edge set as ${\mathcal{E}} = {\mathcal{E}}_1 {\mathbin {\dot{\cup}}} {\mathcal{E}}_2 \ldots {\mathbin {\dot{\cup}}} {\mathcal{E}}_r$ . A matching is a set of (a, p) pairs, $a \in {\mathcal{A}}, p\in{\mathcal{P}}$ such that each a and each p appears in at most one pair. We first consider the popular matching problem; an $O(m\sqrt{n})$ algorithm to solve the popular matching problem was given in [3], where n is the number of vertices and m is the number of edges in the graph. Here we present an O(nω) randomized algorithm for this problem, where ω< 2.376 is the exponent of matrix multiplication. We next consider the rank-maximal matching problem; an $O(\min(mn,Cm\sqrt{n}))$ algorithm was given in [7] for this problem. Here we give an O(Cnω) randomized algorithm, where C is the largest rank of an edge used in such a matching. We also consider a generalization of this problem, called the weighted rank-maximal matching problem, where vertices in ${\mathcal{A}}$ have positive weights

Efficient algorithms for bipartite matching problems with preferences

Matching problems involve a set of participants, where each participant has a capacity and a subset of the participants rank a subset of the others in order of preference (strictly or with ties). Matching problems are motivated in practice by large-scale applications, such as automated matching schemes, which assign participants together based on their preferences over one another. This thesis focuses on bipartite matching problems in which there are two disjoint sets of participants (such as medical students and hospitals). We present a range of efficient algorithms for finding various types of optimal matchings in the context of these problems. Our optimality criteria involve a diverse range of concepts that are alternatives to classical stability. Examples include so-called popular and Pareto optimal matchings, and also matchings that are optimal with respect to their profile (the number of participants obtaining their first choice, second choice and so on). The first optimality criterion that we study is the notion of a Pareto optimal matching, a criterion that economists regard as a fundamental property to be satisfied by an optimal matching. We present the first algorithmic results on Pareto optimality for the Capacitated House Allocation problem (CHA), which is a many-to-one variant of the classical House Allocation problem, as well as for the Hospitals-Residents problem (HR), a generalisation of the classical Stable Marriage problem. For each of these problems, we obtain a characterisation of Pareto optimal matchings, and then use this to obtain a polynomial-time algorithm for finding a maximum Pareto optimal matching. The next optimality criterion that we study is the notion of a popular matching. We study popular matchings in CHA and present a polynomial-time algorithm for finding a maximum popular matching or reporting that none exists, given any instance of CHA. We extend our findings to the case in CHA where preferences may contain ties (CHAT) by proving the extension of a well-known result in matching theory to the capacitated bipartite graph case, and using this to obtain a polynomial-time algorithm for finding a maximum popular matching, or reporting that none exists. We next study popular matchings in the Weighted Capacitated House Allocation problem (WCHA), which is a variant of CHA where the agents have weights assigned to them. We identify a structure in the underlying graph of the problem that singles out those edges that cannot belong to a popular matching. We then use this to construct a polynomial-time algorithm for finding a maximum popular matching or reporting that none exists, for the case where preferences are strict. We then study popular matchings in a variant of the classical Stable Marriage problem with Ties and Incomplete preference lists (SMTI), where preference lists are symmetric. Here, we provide the first characterisation results on popular matchings in the bipartite setting where preferences are two-sided, which can either lead to a polynomial-time algorithm for solving the problem or help establish that it is NP-complete. We also provide the first algorithm for testing if a matching is popular in such a setting. The remaining optimality criteria that we study involve profile-based optimal matchings. We define three versions of what it means for a matching to be optimal based on its profile, namely so-called greedy maximum, rank-maximal and generous maximum matchings. We study each of these in the context of CHAT and the Hospitals-Residents problem with Ties (HRT). For each problem model, we give polynomial-time algorithms for finding a greedy maximum, a rank-maximal and a generous maximum matching.EThOS - Electronic Theses Online ServiceGBUnited Kingdo