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DOI: 10.4230/LIPIcs.FSTTCS.2022.19
A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications

Authors: Rohith Reddy Gangam, Tung Mai, Nitya Raju, and Vijay V. Vazirani

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
Recently [Mai and Vazirani, 2018] identified and initiated work on a new problem, namely understanding structural relationships between the lattices of solutions of two "nearby" instances of stable matching. They also gave an application of their work to finding a robust stable matching. However, the types of changes they allowed in going from instance A to B were very restricted, namely any one agent executes an upward shift. In this paper, we allow any one agent to permute its preference list arbitrarily. Let M_A and M_B be the sets of stable matchings of the resulting pair of instances A and B, and let ℒ_A and ℒ_B be the corresponding lattices of stable matchings. We prove that the matchings in M_A ∩ M_B form a sublattice of both ℒ_A and ℒ_B and those in M_A ⧵ M_B form a join semi-sublattice. These properties enable us to obtain a polynomial time algorithm for not only finding a stable matching in M_A ∩ M_B, but also for obtaining the partial order, as promised by Birkhoff’s Representation Theorem [Birkhoff, 1937]. As a result, we can generate all matchings in this sublattice. Our algorithm also helps solve a version of the robust stable matching problem. We discuss another potential application, namely obtaining new insights into the incentive compatibility properties of the Gale-Shapley Deferred Acceptance Algorithm.
Cite as

Rohith Reddy Gangam, Tung Mai, Nitya Raju, and Vijay V. Vazirani. A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gangam_et_al:LIPIcs.FSTTCS.2022.19,
  author =	{Gangam, Rohith Reddy and Mai, Tung and Raju, Nitya and Vazirani, Vijay V.},
  title =	{{A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.19},
  URN =		{urn:nbn:de:0030-drops-174114},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.19},
  annote =	{Keywords: stable matching, robust solutions, finite distributive lattice, Birkhoff’s Representation Theorem}
}
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