Title:

Efficient algorithms for optimal matching problems under preferences

In this thesis we consider efficient algorithms for matching problems involving preferences, i.e., problems where agents may be required to list other agents that they find acceptable in order of preference. In particular we mainly study the Stable Marriage problem (SM), the Hospitals / Residents problem (HR) and the Student / Project Allocation problem (SPA), and some of their variants. In some of these problems the aim is to find a stable matching which is one that admits no blocking pair. A blocking pair with respect to a matching is a pair of agents that prefer to be matched to each other than their assigned partners in the matching if any. We present an Integer Programming (IP) model for the Hospitals / Residents problem with Ties (HRT) and use it to find a maximum cardinality stable matching. We also present results from an empirical evaluation of our model which show it to be scalable with respect to realworld HRT instance sizes. Motivated by the observation that not all blocking pairs that exist in theory will lead to a matching being undermined in practice, we investigate a relaxed stability criterion called social stability where only pairs of agents with a social relationship have the ability to undermine a matching. This stability concept is studied in instances of the Stable Marriage problem with Incomplete lists (smi) and in instances of hr. We show that, in the smi and hr contexts, socially stable matchings can be of varying sizes and the problem of finding a maximum socially stable matching (max smiss and max hrss respectively) is NPhard though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem arising from restrictions on the social network graph and the lengths of agents’ preference lists. We also consider other optimality criteria with respect to social stability and establish inapproximability bounds for the problems of finding an egalitarian, minimum regret and sex equal socially stable matching in the sm context. We extend our study of social stability by considering other variants and restrictions of max smiss and max hrss. We present NPhardness results for max smiss even under certain restrictions on the degree and structure of the social network graph as well as the presence of master lists. Other NPhardness results presented relate to the problem of determining whether a given manwoman pair belongs to a socially stable matching and the problem of determining whether a given man (or woman) is part of at least one socially stable matching. We also consider the Stable Roommates problem with Incomplete lists under Social Stability (a nonbipartite generalisation of smi under social stability). We observe that the problem of finding a maximum socially stable matching in this context is also NPhard. We present efficient algorithms for three special cases of the problem arising from restrictions on the social network graph and the lengths of agents’ preference lists. These are the cases where (i) there exists a constant number of acquainted pairs (ii) or a constant number of unacquainted pairs or (iii) each preference list is of length at most 2. We also present algorithmic results for finding matchings in the spa context that are optimal with respect to profile, which is the vector whose ith component is the number of students assigned to their ithchoice project. We present an efficient algorithm for finding a greedy maximum matching in the spa context — this is a maximum matching whose profile is lexicographically maximum. We then show how to adapt this algorithm to find a generous maximum matching — this is a matching whose reverse profile is lexicographically minimum. We demonstrate how this approach can allow additional constraints, such as lecturer lower quotas, to be handled flexibly. We also present results of empirical evaluations carried out on both real world and randomly generated datasets. These results demonstrate the scalability of our algorithms as well as some interesting properties of these profilebased optimality criteria. Practical applications of spa motivate the investigation of certain special cases of the problem. For instance, it is often desired that the workload on lecturers is evenly distributed (i.e. load balanced). We enforce this by either adding lower quota constraints on the lecturers (which leads to the potential for infeasible problem instances) or adding a load balancing optimisation criterion. We present efficient algorithms in both cases. Another consideration is the fact that certain projects may require a minimum number of students to become viable. This can be handled by enforcing lower quota constraints on the projects (which also leads to the possibility of infeasible problem instances). A technique of handling this infeasibility is the idea of closing projects that do not meet their lower quotas (i.e. leaving such project completely unassigned). We show that the problem of finding a maximum matching subject to project lower quotas where projects can be closed is NPhard even under severe restrictions on preference lists lengths and project upper and lower quotas. To offset this hardness, we present polynomial time heuristics that find large feasible matchings in practice. We also present ip models for the spa variants discussed and show results obtained from an empirical evaluation carried out on both real and randomly generated datasets. These results show that our algorithms and heuristics are scalable and provide good matchings with respect to profilebased optimality.
