Abstract
In 1976, Knuth asked whether the worst-case running-time of the Gale-Shapley algorithm for the Stable Marriage Problem can be improved when non-sequential access to the input is allowed. Partial negative answers were given by Ng and Hirschberg and as part of Segal's general communication-complexity analysis. We give a far simpler, yet significantly more powerful, argument showing that Ω(n2) Boolean queries of any type are required for finding a stable — or even approximately stable — marriage. Unlike Segal's lower bound, our lower bound generalizes additionally to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's and men's respective preferences profiles, (C) related problems, e.g. whether a given pair is married in every/some stable marriage, (D) whether a proposed marriage is stable or far from stable. To analyze “approximately stable” marriages, we introduce the notion of “distance to stability” and provide an efficient algorithm for its computation.
| Original language | American English |
|---|---|
| Pages (from-to) | 626-647 |
| Number of pages | 22 |
| Journal | Games and Economic Behavior |
| Volume | 118 |
| DOIs | |
| State | Published - Nov 2019 |
Bibliographical note
Funding Information:The work of Rafail Ostrovsky was supported in part by NSF grants 09165174 , 1065276 , 1118126 , 1136174 and 1619348 ; US-Israel BSF grant 2012366 , The Okawa Foundation Research Award, IBM Faculty Research Award, Xerox Faculty Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. This material is based upon work supported in part by DARPA SafeWare program. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government.
Funding Information:
The work of Noam Nisan was supported in part by ISF grants 230/10 and 1435/14 administered by the Israeli Academy of Sciences, and by Israel-USA Bi-national Science Foundation ( BSF ) grant number 2014389 .
Funding Information:
Yannai Gonczarowski is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. The work of Yannai Gonczarowski was supported in part by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. [249159]. The work of Noam Nisan was supported in part by ISF grants 230/10 and 1435/14 administered by the Israeli Academy of Sciences, and by Israel-USA Bi-national Science Foundation (BSF) grant number 2014389. The work of Rafail Ostrovsky was supported in part by NSF grants 09165174, 1065276, 1118126, 1136174 and 1619348; US-Israel BSF grant 2012366, The Okawa Foundation Research Award, IBM Faculty Research Award, Xerox Faculty Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. This material is based upon work supported in part by DARPA SafeWare program. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. We would like to thank the editor and the anonymous referees for many helpful comments.
Funding Information:
Yannai Gonczarowski is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities . The work of Yannai Gonczarowski was supported in part by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. [ 249159 ].
Publisher Copyright:
© 2019 Elsevier Inc.
Keywords
- Approximately stable
- Communication complexity
- Distance to stability
- Stable marriage
- Stable matching