The Geometry of Fractional Stable Matchings and Its Applications

We study the classical stable marriage and stable roommates problems using a polyhedral approach. We propose a new LP formulation for the stable roommates problem, which has a feasible solution if and only if the underlying roommates problem has a stable matching. Furthermore, for certain special weight functions on the edges, we construct a 2-approximation algorithm for the optimal stable roommates problem. Our technique exploits features of the geometry of fractional solutions of this formulation. For the stable marriage problem, we show that a related geometry allows us to express any fractional solution in the stable marriage polytope as a convex combination of stable marriage solutions. This also leads to a genuinely simple proof of the integrality of the stable marriage polytope.

[1]  Tomás Feder,et al.  A New Fixed Point Approach for Stable Networks and Stable Marriages , 1992, J. Comput. Syst. Sci..

[2]  Uriel G. Rothblum,et al.  Canonical monotone decompositions of fractional stable matchings , 1996 .

[3]  Uriel G. Rothblum,et al.  Characterization of stable matchings as extreme points of a polytope , 1992, Math. Program..

[4]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[5]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[6]  J. V. Vate.  Linear programming brings marital bliss , 1989 .

[7]  Richard M. Karp,et al.  On linear characterizations of combinatorial optimization problems , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[8]  W IrvingRobert,et al.  An efficient algorithm for the optimal stable marriage , 1987 .

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

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

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

[12]  Robert W. Irving.  An Efficient Algorithm for the "Stable Roommates" Problem , 1985, J. Algorithms.

[13]  Ashok Subramanian,et al.  A New Approach to Stable Matching Problems , 1989, SIAM J. Comput..

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

[15]  Uriel G. Rothblum,et al.  Stable Matchings and Linear Inequalities , 1994, Discret. Appl. Math..

[16]  Alvin E. Roth,et al.  Stable Matchings, Optimal Assignments, and Linear Programming , 1993, Math. Oper. Res..

[17]  Chung-Piaw Teo,et al.  LP based approach to optimal stable matchings , 1997, SODA '97.