Stable flows with restricted edges

The stable marriage problem with its extensions is a widely studied subject. In this paper, we combine two topics related to it, setting up new and generalizing known results in both. The stable flow problem extends the well-known stable matching problem to network flows. Restricted edges have some special properties: forced edges must be in the stable solution, while forbidden edges may not be in it. Free edges are not able to block matchings. Here we describe a polynomial algorithm that finds a stable flow with forced and forbidden edges or proves its nonexistence. In contrast to this, we also show that determining whether a stable flow with free edges exists, is NP-complete.

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

[2]  Laurent Viennot,et al.  Acyclic Preference Systems in P2P Networks , 2007, Euro-Par.

[3]  Brian C. Dean,et al.  Faster Algorithms for Stable Allocation Problems , 2010, Algorithmica.

[4]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[5]  Tamás Király,et al.  A note on kernels and Sperner's Lemma , 2009, Discret. Appl. Math..

[6]  Uriel G. Rothblum,et al.  A stable matching model with an entrance criterion applied to the assignment of students to dormitories at the technion , 2008, Int. J. Game Theory.

[7]  David Manlove,et al.  Socially Stable Matchings in the Hospitals/Residents Problem , 2013, WADS.

[8]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[9]  M. Balinski,et al.  A Tale of Two Mechanisms: Student Placement , 1999 .

[10]  Martin Skutella,et al.  Stable Flows over Time , 2013, Algorithms.

[11]  Alvin E. Roth,et al.  Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis , 1990 .

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

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

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

[15]  M. Ostrovsky.  Stability in Supply Chain Networks , 2005 .

[16]  David Manlove,et al.  Efficient algorithms for generalized Stable Marriage and Roommates problems , 2007, Theor. Comput. Sci..

[17]  Celina M. H. de Figueiredo,et al.  The stable marriage problem with restricted pairs , 2003, Theor. Comput. Sci..

[18]  Tamás Fleiner.  On Stable Matchings and Flows , 2010, WG.

[19]  Michel Balinski,et al.  Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry) , 2000, Discret. Appl. Math..

[20]  Tamás Fleiner,et al.  Egerváry Research Group on Combinatorial Optimization on the Stable B-matching Polytope on the Stable B-matching Polytope , 2022 .

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

[22]  Tamás Fleiner,et al.  Stable roommates with free edges Kataŕına Cechlárová and Tamás Fleiner , 2009 .

[23]  Dorothea Kübler,et al.  Telling the Truth May Not Pay Off : An Empirical Study of Centralized University Admissions in Germany , 2011 .