A Stable Marriage Requires Communication

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take $\Theta(n^2)$ steps to find a stable marriage in the worst case, but only $\Theta(n \log n)$ steps in the average case (with $n$ women and $n$ men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that $\Theta(n^2)$ queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general - albeit slightly weaker - lower bound follows from Segal's general analysis of communication complexity, namely that $\Omega(n^2)$ Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries. Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that $\Omega(n^2)$ Boolean queries of any type are indeed required for finding a stable - or even an approximately stable - marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation.

[1]  Alvin E. Roth,et al.  Unraveling Yields Inefficient Matchings: Evidence from Post-Season College Football Bowls , 2004 .

[2]  Ehud Friedgut,et al.  Sisterhood in the Gale-Shapley Matching Algorithm , 2011, Electron. J. Comb..

[3]  Noam Nisan,et al.  Economic efficiency requires interaction , 2013, STOC.

[4]  Jacob D. Leshno,et al.  Unbalanced Random Matching Markets: The Stark Effect of Competition , 2017, Journal of Political Economy.

[5]  J. Picard.  Maximal Closure of a Graph and Applications to Combinatorial Problems , 1976 .

[6]  Dan Gusfield,et al.  Three Fast Algorithms for Four Problems in Stable Marriage , 1987, SIAM J. Comput..

[7]  Hao Li,et al.  Unraveling in Matching Markets , 1998 .

[8]  A. Roth,et al.  Unraveling Reduces Mobility in a Labor Market: Gastroenterology with and without a Centralized Match , 2003, Journal of Political Economy.

[9]  Robert W. Irving,et al.  The Complexity of Counting Stable Marriages , 1986, SIAM J. Comput..

[10]  Alexander A. Razborov,et al.  On the Distributional Complexity of Disjointness , 1992, Theor. Comput. Sci..

[11]  Shengwu Li.  Obviously Strategy-Proof Mechanisms , 2017 .

[12]  A. Roth,et al.  Jumping the Gun: Imperfections and Institutions Related to the Timing of Market Transactions , 1994 .

[13]  Bala Kalyanasundaram,et al.  The Probabilistic Communication Complexity of Set Intersection , 1992, SIAM J. Discret. Math..

[14]  A. Roth.  A natural experiment in the organization of entry-level labor markets: regional markets for new physicians and surgeons in the United Kingdom. , 1991, The American economic review.

[15]  L. B. Wilson,et al.  The stable marriage problem , 1971, Commun. ACM.

[16]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[17]  Dorit S. Hochbaum.  A new - old algorithm for minimum-cut and maximum-flow in closure graphs , 2001, Networks.

[18]  L. B. Wilson.  An analysis of the stable marriage assignment algorithm , 1972 .

[19]  Rafail Ostrovsky,et al.  Fast Distributed Almost Stable Matchings , 2015, PODC.

[20]  Olle Häggström,et al.  Instability of matchings in decentralized markets with various preference structures , 2008, Int. J. Game Theory.

[21]  M. Utku Ünver.  On the survival of some unstable two-sided matching mechanisms , 2005, Int. J. Game Theory.

[22]  Yannai A. Gonczarowski,et al.  No Stable Matching Mechanism is Obviously Strategy-Proof , 2015 .

[23]  Harry R. Lewis,et al.  Review of "Mariages stables et leur relations avec d'autre problèmes combinatoires: introduction à l'analyze mathématique des algorithmes" by Donald E. Knuth. Les Presses de l'Université de Montréal. , 1978, SIGA.

[24]  L. S. Shapley,et al.  College Admissions and the Stability of Marriage , 2013, Am. Math. Mon..

[25]  Robert W. Irving,et al.  The Stable marriage problem - structure and algorithms , 1989, Foundations of computing series.

[26]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[27]  Dana Randall,et al.  Sampling stable marriages: why spouse-swapping won't work , 2008, SODA '08.

[28]  Rafail Ostrovsky,et al.  Fast distributed almost stable marriages , 2014, ArXiv.

[29]  Yannai A. Gonczarowski.  Manipulation of stable matchings using minimal blacklists , 2013, EC.

[30]  Ilya Segal,et al.  The communication requirements of social choice rules and supporting budget sets , 2007, J. Econ. Theory.

[31]  David A. Freedman,et al.  Machiavelli and the Gale-Shapley Algorithm , 1981 .

[32]  A. Roth.  The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory , 1984, Journal of Political Economy.

[33]  Chi-Jen Lu,et al.  Communication Requirements for Stable Marriages , 2010, CIAC.

[34]  Jacob D. Leshno,et al.  Unbalanced random matching markets , 2013, EC '13.

[35]  Martin Dyer,et al.  The Stable Marriage Problem: Structure and Algorithms , 1991 .

[36]  A. Roth.  On the Allocation of Residents to Rural Hospitals: A General Property of Two-Sided Matching Markets , 1986 .

[37]  Ning Chen,et al.  On the complexity of trial and error , 2012, STOC '13.

[38]  Daniel S. Hirschberg,et al.  Lower Bounds for the Stable Marriage Problem and its Variants , 1990, SIAM J. Comput..