Abstract
The aim of this note is to point out some combinatorial applications of a lemma of Scarf, proved first in the context of game theory. The usefulness of the lemma in combinatorics has already been demonstrated in [1], where it was used to prove the existence of fractional kernels in digraphs not containing cyclic triangles. We indicate some links of the lemma to other combinatorial results, both in terms of its statement (being a relative of the Gale-Shapley theorem) and its proof (in which respect it is a kin of Sperner’s lemma). We use the lemma to prove a fractional version of the Gale-Shapley theorem for hypergraphs, which in turn directly implies an extension of this theorem to general (not necessarily bipartite) graphs due to Tan [12]. We also prove the following result, related to a theorem of Sands, Sauer and Woodrow [10]: given a family of partial orders on the same ground set, there exists a system of weights on the vertices, which is (fractionally) independent in all orders, and each vertex is dominated by them in one of the orders.
Research supported by the fund for the promotion of research at the Technion and by a grant from the Israel Science Foundation.
Part of the research was done at the Centrum voor Wiskunde en Informatica (CWI) and during two visits to the Technion, Haifa. Research was supported by the Netherlands Organization for Scientific Research (NWO) and by OTKA T 029772.
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References
R. Aharoni and R. Holzman. Fractional kernels in digraphs. J. Combin. Theory Ser. B, 73(1):1–6, 1998.
J. Edmonds. Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), pages 69–87. Gordon and Breach, New York, 1970.
T. Fleiner. A fixed-point approach to stable matchings and some applications. submitted to Mathematics of Operations Research and EGRES Technical Report TR-2001-01, http://www.cs.elte.hu/egres, 2001 March
T. Fleiner. Stable and crossing structures, August, 2000. PhD dissertation, http://www.renyi.hu/~fleiner.
D. Gale and L.S. Shapley. College admissions and stability of marriage. Amer. Math. Monthly, 69(1):9–15, 1962.
R. W. Irving. An efficient algorithm for the “stable roommates” problem. J. Algorithms, 6(4):577–595, 1985.
L. Lovász, L. Matroids and Sperner’s lemma, European J. Combin. 1 (1980), no. 1, 65–66.
L. Lovász and M. D. Plummer. Matching theory. North-Holland Publishing Co., Amsterdam, 1986. Annals of Discrete Mathematics, 29.
L.S Shapley, On balanced games without side payments, in T.C. Hu and S.M. Robinson, eds., Mathematical Programming, Academic Press, NY (1973), 261–290.
B. Sands, N. Sauer, and R. Woodrow. On monochromatic paths in edge-coloured digraphs. J. Combin. Theory Ser. B, 33(3):271–275, 1982.
H. E. Scarf. The core of an N person game. Econometrica, 35:50–69, 1967.
J. J.M. Tan. A necessary and sufficient condition for the existence of a complete stable matching. J. Algorithms, 12(1):154–178, 1991.
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Aharoni, R., Fleiner, T. (2002). On a Lemma of Scarf. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. IPCO 2002. Lecture Notes in Computer Science, vol 2337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47867-1_13
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DOI: https://doi.org/10.1007/3-540-47867-1_13
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