On Treewidth and Stable Marriage

Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESM), Balanced Stable Marriage (BSM), max-Stable Marriage with Ties (max-SMT) and min-Stable Marriage with Ties (min-SMT) problems. In this paper, we analyze these problems from the viewpoint of Parameterized Complexity. We conduct the first study of these problems with respect to the parameter treewidth. First, we study the treewidth $\mathtt{tw}$ of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time $n^{O(\mathtt{tw})}$, we prove that all of these algorithms are likely to be essentially optimal. Next, we study the treewidth $\mathtt{tw}$ of the rotation digraph. In this context, the max-SMT and min-SMT are not defined. For both SESM and BSM, we design (non-trivial) algorithms that run in time $2^{\mathtt{tw}}n^{O(1)}$. Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time $(2-\epsilon)^{\mathtt{tw}}n^{O(1)}$ do not exist for any fixed $\epsilon>0$. We thus present a comprehensive, complete picture of the behavior of central optimization versions of Stable Marriage with respect to treewidth.

[1]  Zoltán Király,et al.  Better and Simpler Approximation Algorithms for the Stable Marriage Problem , 2008, Algorithmica.

[2]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[3]  Patrick Prosser,et al.  An Empirical Study of the Stable Marriage Problem with Ties and Incomplete Lists , 2002, ECAI.

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

[5]  Hiroki Yanagisawa.  Approximation algorithms for stable marriage problems , 2007 .

[6]  Donald E. Knuth,et al.  Stable Networks and Product Graphs , 1995 .

[7]  Akiko Kato,et al.  Complexity of the sex-equal stable marriage problem , 1993 .

[8]  Donald E. Knuth,et al.  Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms , 1996 .

[9]  K. Iwama,et al.  A Survey of the Stable Marriage Problem and Its Variants , 2008, International Conference on Informatics Education and Research for Knowledge-Circulating Society (icks 2008).

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

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

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

[13]  David Manlove,et al.  Approximability results for stable marriage problems with ties , 2003, Theor. Comput. Sci..

[14]  Yoshio Okamoto,et al.  On Problems as Hard as CNF-SAT , 2011, 2012 IEEE 27th Conference on Computational Complexity.

[15]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[16]  Michael R. Fellows,et al.  Review of: Fundamentals of Parameterized Complexity by Rodney G. Downey and Michael R. Fellows , 2015, SIGA.

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

[18]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

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

[20]  Dániel Marx,et al.  Parameterized Complexity and Local Search Approaches for the Stable Marriage Problem with Ties , 2009, Algorithmica.

[21]  Robert W. Irving,et al.  An efficient algorithm for the “optimal” stable marriage , 1987, JACM.

[22]  Russell Impagliazzo,et al.  A duality between clause width and clause density for SAT , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[23]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

[24]  Eric McDermid,et al.  Sex-Equal Stable Matchings: Complexity and Exact Algorithms , 2012, Algorithmica.

[25]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[26]  Dimitrios Tsoumakos,et al.  An Equitable Solution to the Stable Marriage Problem , 2015, 2015 IEEE 27th International Conference on Tools with Artificial Intelligence (ICTAI).

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

[28]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[29]  Francesca Rossi,et al.  Solving Hard Stable Matching Problems via Local Search and Cooperative Parallelization , 2015, AAAI.

[30]  Shuichi Miyazaki,et al.  A 25/17-Approximation Algorithm for the Stable Marriage Problem with One-Sided Ties , 2010, ESA.

[31]  Russell Impagliazzo,et al.  On the Complexity of k-SAT , 2001, J. Comput. Syst. Sci..

[32]  Hyun Kim,et al.  Ant Colony based Algorithm for Stable Marriage Problem , 2007 .

[33]  Eric McDermid,et al.  A structural approach to matching problems with preferences , 2011 .

[34]  David Manlove,et al.  Algorithmics of Matching Under Preferences , 2013, Bull. EATCS.

[35]  Michael R. Fellows,et al.  On the parameterized complexity of multiple-interval graph problems , 2009, Theor. Comput. Sci..

[36]  David Manlove,et al.  The structure of stable marriage with indifference , 2002, Discret. Appl. Math..

[37]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[38]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[39]  Telikepalli Kavitha,et al.  Improved approximation algorithms for two variants of the stable marriage problem with ties , 2015, Math. Program..

[40]  Gregg O'Malley,et al.  Algorithmic aspects of stable matching problems , 2007 .

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

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

[43]  Seiki Kyan,et al.  Genetic algorithm for sex-fair stable marriage problem , 1995, Proceedings of ISCAS'95 - International Symposium on Circuits and Systems.