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.
Highlights
Stable Matching together with its in numerous variants are among the most well-studied problems in matching theory, driven by applications to economics, business, engineering, and more recently medical sciences
Ever since the theoretical framework for Stable Matching was laid down by Gale and Shapley [9] to study the current heuristic used to assign medical residents to hospitals in New England, the topic has received considerable attention from theoreticians and practitioners alike. It is one of the foundational problems in social choice theory, where a matching is viewed as an allocation or assignment of resources to Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 29:2 Stable Matching Games: Manipulation via Subgraph Isomorphism relevant agents, whereby the nature of the assignment can vary greatly depending on the scenario/marketplace they are modelling
In this article we give a 2O(n) algorithm, which breaks the naïve bound, and uses an idea which connects Stable Extension of Partial Matching (SEOPM) to the problem of Colored Subgraph Isomorphism
Summary
Stable Matching together with its in numerous variants are among the most well-studied problems in matching theory, driven by applications to economics, business, engineering, and more recently medical sciences. Ever since the theoretical framework for Stable Matching was laid down by Gale and Shapley [9] to study the current heuristic used to assign medical residents to hospitals in New England, the topic has received considerable attention from theoreticians and practitioners alike. It is one of the foundational problems in social choice theory, where a matching is viewed as an allocation or assignment of resources to. Given any set of preference lists of men and women there exists at least one stable matching They gave a polynomial time algorithm to find a stable matching. We will use (LM , LW ) to denote the set of preference lists of men and women, and the men-optimal matching with respect to these lists is denoted by GS(LM , LW )
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