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License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2022.19
URN: urn:nbn:de:0030-drops-174114
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17411/
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Gangam, Rohith Reddy ; Mai, Tung ; Raju, Nitya ; Vazirani, Vijay V.

A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications

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LIPIcs-FSTTCS-2022-19.pdf (0.8 MB)

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.

BibTeX - Entry

@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/opus/volltexte/2022/17411},
  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}
}

Keywords: stable matching, robust solutions, finite distributive lattice, Birkhoff’s Representation Theorem
Collection: 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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