Popular Matchings in the Weighted Capacitated House Allocation Problem

We consider the problem of finding a popular matching in the Weighted Capacitated House Allocation problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight indicating his priority, and a preference list in which a subset of houses are ranked in strict order. Each house has a capacity that indicates the maximum number of agents who could be matched to it. A matching M of agents to houses is popular if there is no other matching M′ such that the total weight of the agents who prefer their allocation in M′ to that in M exceeds the total weight of the agents who prefer their allocation in M to that in M′ . Here, we give an O( √ Cn1 + m) algorithm to determine if an instance of WCHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n1 is the number of agents, and m is the total length of the agents’ preference lists

Popular matchings in the weighted capacitated house allocation problem

We consider the problem of finding a popular matching in the <i>Weighted Capacitated House Allocation</i> problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight indicating his priority, and a preference list in which a subset of houses are ranked in strict order. Each house has a capacity that indicates the maximum number of agents who could be matched to it. A matching M of agents to houses is popular if there is no other matching M′ such that the total weight of the agents who prefer their allocation in M′ to that in M exceeds the total weight of the agents who prefer their allocation in M to that in M′. Here, we give an [FORMULA] algorithm to determine if an instance of WCHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n1 is the number of agents, and m is the total length of the agents' preference lists

Counting Popular Matchings in House Allocation Problems

We study the problem of counting the number of popular matchings in a given instance. A popular matching instance consists of agents A and houses H, where each agent ranks a subset of houses according to their preferences. A matching is an assignment of agents to houses. A matching M is more popular than matching M' if the number of agents that prefer M to M' is more than the number of people that prefer M' to M. A matching M is called popular if there exists no matching more popular than M. McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered. We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem. We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard

Popular matchings in the marriage and roommates problems

Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases

On Weakly and Strongly Popular Rankings

Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list of all candidates. A popular ranking pi of the candidates is at least as good as any other ranking sigma in the following sense: if we compare pi to sigma, at least half of all voters will always weakly prefer pi. Whether a voter prefers one ranking to another is calculated based on the Kendall distance. A more traditional definition of popularity---as applied to popular matchings, a well-established topic in computational social choice---is stricter, because it requires at least half of the voters who are not indifferent between pi and sigma to prefer pi. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results by van Zylen et al. We also point out connections to the famous open problem of finding a Kemeny consensus with 3 voters

Minimal Envy and Popular Matchings

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them

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

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