Theory And Algorithms For Matching Problems Under Preferences

Matching under preferences involves matching agents to one another, subject to various optimality criteria such as stability, popularity, and Pareto-optimality, etc. Each agent expresses ordinal preferences over a subset of the others. Real-life applications include assigning graduating medical students to hospitals, high school students to colleges, public houses to applicants, and so on. We consider various matching problems with preferences. In this dissertation, we present efficient algorithms to solve them, prove hardness results, and develop linear programming theory around them. In the first part of this dissertation, we present two characterizations for the set of super-stable matchings. Super-stability is one of the optimality criteria when the preference lists contain ties. The first algorithm computes irreducible super-stable matchings in the super-stable matching lattice. The second algorithm takes O(mn) time, where m denotes the number of edges and n denotes the number of vertices and gives an explicit rotation poset that can be used to construct all super-stable matchings. In the second part, we present a polyhedral characterization of the set of all super-stable matchings, i.e. a linear system that is integral and describes the super-stable matching polytope. We also give alternative proof for the integrality of the strongly stable matching polytope. We also use linear programming techniques to solve an application of the stable matching problem. In the third part, we present NC algorithms for the popular matching problem. Popularity is another optimality criterion, where each agent gives a vote and the outcome matching has majority votes. In the last part, we consider envy-freeness, a relaxation of stability in the Hospitals/Residents setting, which allows blocking pairs involving a resident and an empty position of a hospital. Envy-free matching might not exist. We prove NP-hardness results of minimizing envy (if envy is inevitable) in terms of envy-pairs and envy-residents in the Hospitals/Residents Problem with Lower Quota