Date of Award

8-2007

Document Type

Thesis

Degree Name

Master of Science (MS)

Legacy Department

Computer Science

Advisor

Dean, Brian

Committee Member

Jacobs , David

Committee Member

Wang , James

Abstract

We consider a high-multiplicity generalization of the classical stable matching problem known as the stable allocation problem, introduced by Baiou and Balinski in 2002. By leveraging new structural properties and sophisticated data structures, we show how to solve this problem in O(mlog n) time on an bipartite instance with n nodes and m edges, improving the best known running time of O(mn). Our approach simplifies the algorithmic landscape for this problem by providing a common generalization of two different approaches from the literature -- the classical Gale-Shapley algorithm, and a recent algorithm of Baiou and Balinski. Building on this algorithm, we provide an O(m log n) algorithm for the non-bipartite stable allocation problem that introduces a new and useful transformation from non-bipartite to bipartite instances. We also give a polynomial-time algorithm for solving the 'optimal' variant of the bipartite stable allocation problem, as well as a 2-approximation algorithm for the NP-hard 'optimal' variant of the non-bipartite stable allocation problem. Finally, we highlight some important connections between the stable allocation problem and the maximum flow problem.

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