Finding large stable matchings

When ties and incomplete preference lists are permitted in the stable marriage and hospitals/residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size, and position of ties. In this article, we present two new heuristics for finding large stable matchings in variants of these problems in which ties are on one side only. We describe an empirical study involving these heuristics and the best existing approximation algorithm for this problem. Our results indicate that all three of these algorithms perform significantly better than naive tie-breaking algorithms when applied to real-world and randomly-generated data sets and that one of the new heuristics fares slightly better than the other algorithms, in most cases. This study, and these particular problem variants, are motivated by important applications in large-scale centralized matching schemes.

[1]  Magnús M. Halldórsson,et al.  Improved Approximation of the Stable Marriage Problem , 2003, ESA.

[2]  Shuichi Miyazaki,et al.  A (2 - clog N/N)-Approximation Algorithm for the Stable Marriage Problem , 2006, IEICE Trans. Inf. Syst..

[3]  David Manlove,et al.  Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems , 2008, J. Comb. Optim..

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

[5]  Aytek Erdil,et al.  What's the Matter with Tie-Breaking? Improving Efficiency in School Choice , 2008 .

[6]  Zoltán Király.  Better and Simpler Approximation Algorithms for the Stable Marriage Problem , 2009, Algorithmica.

[7]  Magnús M. Halldórsson,et al.  Randomized Approximation of the Stable Marriage Problem , 2003, COCOON.

[8]  Eric A. Houck.  Student Assignment Policy , 2011 .

[9]  Parag A. Pathak,et al.  Changing the Boston School Choice Mechanism , 2006 .

[10]  Antonio Romero-Medina,et al.  Implementation of stable solutions in a restricted matching market , 1998 .

[11]  Michel Balinski,et al.  Student admissions and faculty recruitment , 2004, Theor. Comput. Sci..

[12]  David Gale,et al.  Some remarks on the stable matching problem , 1985, Discret. Appl. Math..

[13]  Harold N. Gabow,et al.  An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems , 1983, STOC.

[14]  Shuichi Miyazaki,et al.  A (2-c(log N/N))-Approximation Algorithm for the Stable Marriage Problem , 2004, SWAT.

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

[16]  Parag A. Pathak,et al.  The New York City High School Match , 2005 .

[17]  Magnús M. Halldórsson,et al.  Improved approximation results for the stable marriage problem , 2007, TALG.

[18]  David Manlove,et al.  Stable Marriage with Ties and Bounded Length Preference Lists , 2006, ACiD.

[19]  Shuichi Miyazaki,et al.  A 1.875: approximation algorithm for the stable marriage problem , 2006, SODA '07.

[20]  Robert W. Irving.  Stable Marriage and Indifference , 1994, Discret. Appl. Math..

[21]  P. Biró.  Higher education admission in Hungary by a “score-limit algorithm” , 2007 .

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

[23]  David Manlove,et al.  Hard variants of stable marriage , 2002, Theor. Comput. Sci..

[24]  Eric McDermid.  A 3/2-Approximation Algorithm for General Stable Marriage , 2009, ICALP.