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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.8

Stable Matching: Choosing Which Proposals to Make

Authors: Ishan Agarwal and Richard Cole

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

To guarantee all agents are matched in general, the classic Deferred Acceptance algorithm needs complete preference lists. In practice, preference lists are short, yet stable matching still works well. This raises two questions: - Why does it work well? - Which proposals should agents include in their preference lists? We study these questions in a model, introduced by Lee [Lee, 2016], with preferences based on correlated cardinal utilities: these utilities are based on common public ratings of each agent together with individual private adjustments. Lee showed that for suitable utility functions, in large markets, with high probability, for most agents, all stable matchings yield similar valued utilities. By means of a new analysis, we strengthen Lee’s result, showing that in large markets, with high probability, for all but the agents with the lowest public ratings, all stable matchings yield similar valued utilities. We can then deduce that for all but the agents with the lowest public ratings, each agent has an easily identified length O(log n) preference list that includes all of its stable matches, addressing the second question above. We note that this identification uses an initial communication phase. We extend these results to settings where the two sides have unequal numbers of agents, to many-to-one settings, e.g. employers and workers, and we also show the existence of an ε-Bayes-Nash equilibrium in which every agent makes relatively few proposals. These results all rely on a new technique for sidestepping the conditioning between the tentative matching events that occur over the course of a run of the Deferred Acceptance algorithm. We complement these theoretical results with an experimental study.

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Ishan Agarwal and Richard Cole. Stable Matching: Choosing Which Proposals to Make. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agarwal_et_al:LIPIcs.ICALP.2023.8,
  author =	{Agarwal, Ishan and Cole, Richard},
  title =	{{Stable Matching: Choosing Which Proposals to Make}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.8},
  URN =		{urn:nbn:de:0030-drops-180603},
  doi =		{10.4230/LIPIcs.ICALP.2023.8},
  annote =	{Keywords: Stable matching, randomized analysis}
}

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APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.43

Nearly Optimal Embeddings of Flat Tori

Authors: Ishan Agarwal, Oded Regev, and Yi Tang

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract

We show that for any n-dimensional lattice ℒ ⊆ ℝⁿ, the torus ℝⁿ/ℒ can be embedded into Hilbert space with O(√{nlog n}) distortion. This improves the previously best known upper bound of O(n√{log n}) shown by Haviv and Regev (APPROX 2010, J. Topol. Anal. 2013) and approaches the lower bound of Ω(√n) due to Khot and Naor (FOCS 2005, Math. Ann. 2006).

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Ishan Agarwal, Oded Regev, and Yi Tang. Nearly Optimal Embeddings of Flat Tori. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agarwal_et_al:LIPIcs.APPROX/RANDOM.2020.43,
  author =	{Agarwal, Ishan and Regev, Oded and Tang, Yi},
  title =	{{Nearly Optimal Embeddings of Flat Tori}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{43:1--43:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.43},
  URN =		{urn:nbn:de:0030-drops-126464},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.43},
  annote =	{Keywords: Lattices, metric embeddings, flat torus}
}

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Stable Matching: Choosing Which Proposals to Make

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Nearly Optimal Embeddings of Flat Tori

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