Kernels, Stable Matchings, and Scarf's Lemma (Combinatorial Optimization and Discrete Algorithms)

Scarf’s Lemma originally appeared as a tool to prove the non‐emptiness of the core of certain NTU games. More recently, however, several applications have been found in the area of graph theory and discrete mathematics. In this paper we present and extend some of these applications. In particular, we prove results on the existence of kernels in orientations of h‐ perfect graphs. We describe a new direct link between Scarf’s Lemma and Sperner’s Lemma giving a new proof to the former.

[1]  C. Rioux.  Scarf's Theorem and Applications in Combinatorics , 2006 .

[2]  Tamás Fleiner,et al.  On a lemma of Scarf , 2002, J. Comb. Theory, Ser. B.

[3]  Bert Gerards,et al.  Matrices with the edmonds—Johnson property , 1986, Comb..

[4]  Vladimir Gurvich,et al.  Perfect graphs, kernels, and cores of cooperative games , 2006, Discret. Math..

[5]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[6]  H. Scarf.  The Core of an N Person Game , 1967 .

[7]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computation , 2010, 50 Years of Integer Programming.

[8]  F. Bruce Shepherd,et al.  Applying Lehman's theorems to packing problems , 1995, Math. Program..

[9]  Vladimir Gurvich,et al.  Perfect graphs are kernel solvable , 1996, Discret. Math..

[10]  Ron Holzman,et al.  Fractional Kernels in Digraphs , 1998, J. Comb. Theory, Ser. B.

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

[12]  Ali Ridha Mahjoub,et al.  Compositions of Graphs and Polyhedra III: Graphs with No W4 Minor , 1994, SIAM J. Discret. Math..

[13]  George L. Nemhauser,et al.  Polyhedral Characterizations, Perfection of Line Graphs , 1998, Discret. Appl. Math..

[14]  J. P. Uhry,et al.  A class of h-perfect graphs , 1984, Discret. Math..

[15]  V. Gurvich,et al.  Stable families of coalitions and normal hypergraphs , 1997 .

[16]  Jimmy J. M. Tan.  A Necessary and Sufficient Condition for the Existence of a Complete Stable Matching , 1991, J. Algorithms.

[17]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..