A Study Of Stable Marriage Problems With Ties

We study a number of variants of the Stable Marriage problem. Such problems have a long history, but there has been an upsurge in interest in recent years as the various possibilities that arise when ties are allowed in preference lists have been studied. The inclusion of ties in preference lists gives rise to three different versions of stability, so-called super-stability, strong stability and weak stability. The study of these three versions has thrown up a number of challenging problems, and this thesis contributes a range of new results in some of these areas. The first variant that we study is the Hospitals/Residents problem with ties, a manyto-one variant of the Stable Marriage problem. We present two different polynomial-time algorithms for finding a strongly stable matching, one favouring the residents and the other the hospitals. We also study the Stable Roommates problem with Ties, a non-bipartite variant of the Stable Marriage problem, and we again present a polynomial-time algorithm for finding a strongly stable matching. We then introduce the Stable Fixtures problem, a many-to-many variant of the Stable Roommates problem, initially focusing on the version in which preferences are strict. We present an algorithm, which runs in time linear in the input size, to find a stable matching. We then consider the problem when ties are allowed in the preference lists, and we present an algorithm, again linear in the input size, to find a super-stable matching. We also study the structure underlying the set of super-stable matchings for the Stable Marriage problem with ties (SMT). We extend the concept of a rotation, essentially the minimum difference between stable matchings, to super-stability, and show that we can construct a directed acyclic graph to represent precedence amongst these meta-rotations. We then use this structure to show that we can find an egalitarian super-stable matching, a minimum regret super-stable matching and all the super-stable pairs in polynomialtime, and generate all the super-stable matchings with polynomial-time between successive matchings. We then explore some of the issues arising in weak stability, where a number of the key problems are known to be NP-hard. We show a relationship between the sizes of weakly stable matchings and the size of a strongly stable matching in an instance of SMT, and also in a number of the other variants. We give an improved approximation algorithm

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