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DOI: 10.4230/LIPIcs.FSTTCS.2022.19
URN: urn:nbn:de:0030-drops-174114
URL: https://drops.dagstuhl.de/opus/volltexte/2022/17411/
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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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 | |
| Seminar: | 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022) | |
| Issue date: | 2022 | |
| Date of publication: | 14.12.2022 |
