Abstract
In this paper we consider a problem that arises from a strategic issue in the stable matching model (with complete preference lists) from the viewpoint of exact-exponential time algorithms. Specifically, we study the Stable Extension of Partial Matching (SEOPM) problem, where the input consists of the complete preference lists of men, and a partial matching. The objective is to find (if one exists) a set of preference lists of women, such that the men-optimal Gale–Shapley algorithm outputs a perfect matching that contains the given partial matching. Kobayashi and Matsui (Algorithmica 58:151–169, 2010) proved this problem is NP-complete. In this article, we give an exact-exponential algorithm for SEOPM running in time \(2^{{\mathcal {O}} (n)}\), where n denotes the number of men/women. We complement our algorithmic finding by showing that unless Exponential Time Hypothesis (ETH) fails, our algorithm is asymptotically optimal. That is, unless ETH fails, there is no algorithm for SEOPM running in time \(2^{o(n)}\). Our algorithm is a non-trivial combination of a parameterized algorithm for Subgraph Isomorphism, a relationship between stable matching and finding an out-branching in an appropriate graph and enumerating all possible non-isomorphic out-branchings. Our results cover both the cases when the preference lists are strict and complete, and when they are strict but possibly incomplete.
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Notes
In the stable roommates problem, the matching market consists of agents of the same type, as opposed to the market modeled the stable marriage problem that consists of agents of two types, men and women. Roommates assignments in college housing facilities is a real world application of the stable roommates problem.
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Celebrating the 60th Birthday of Gregory Gutin.
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Gupta, S., Roy, S. Stable Matching Games: Manipulation via Subgraph Isomorphism. Algorithmica 80, 2551–2573 (2018). https://doi.org/10.1007/s00453-017-0382-5
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DOI: https://doi.org/10.1007/s00453-017-0382-5