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Lower Bounds for Partial Matchings in Regular Bipartite Graphs and Applications to the Monomer–Dimer Entropy

Published online by Cambridge University Press:  01 May 2008

SHMUEL FRIEDLAND and
LEONID GURVITS
SHMUEL FRIEDLAND
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA (e-mail: friedlan@uic.edu)
LEONID GURVITS
Affiliation:
Los Alamos National Laboratories, Los Alamos, NM 87545, USA (e-mail: gurvits@lanl.gov)
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Abstract

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We derive here the Friedland–Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in r-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three-dimensional monomer–dimer entropy.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008, pp. 347 - 361
Copyright
Copyright © Cambridge University Press 2007

References

[1]Baxter, R. J. (1968) Dimers on a rectangular lattice. J. Math. Phys. 9 650–654.CrossRefGoogle Scholar
[2]Bregman, L. M. (1973) Some properties of nonnegative matrices and their permanents. Soviet Math. Dokl. 14 945–949.Google Scholar
[3]Egorichev, G. P. (1981) Proof of the van der Waerden conjecture for permanents. Siberian Math. J. 22 854–859.Google Scholar
[4]Erdős, P. and Rényi, A. (1968) On random matrices II. Studia Math. Hungar. 3 459–464.Google Scholar
[5]Falikman, D. I. (1981) Proof of the van der Waerden conjecture regarding the permanent of doubly stochastic matrix. Math. Notes Acad. Sci. USSR 29 475–479.Google Scholar
[6]Friedland, S. (1979) A lower bound for the permanent of doubly stochastic matrices. Ann. of Math. 110 167–176.CrossRefGoogle Scholar
[7]Friedland, S. (1982) A proof of a generalized van der Waerden conjecture on permanents. Linear Multilinear Algebra 11 107–120.CrossRefGoogle Scholar
[8]Friedland, S. and Gurvits, L. (2006) Generalized Friedland-Tverberg inequality: Applications and extensions. arXiv:math/0603410 v2.Google Scholar
[9]Friedland, S., Krop, E., Lundow, P. H. and Markström, K. (2007) Validations of the asymptotic matching conjectures. arXiv:math.CO/0603001 v1.Google Scholar
[10]Friedland, S., Krop, E. and Markström, K. Enumerating matchings in regular graphs, in preparation.Google Scholar
[11]Friedland, S. and Peled, U. N. (2005) Theory of computation of multidimensional entropy with an application to the monomer-dimer problem. Adv. Appl. Math. 34 486–522.CrossRefGoogle Scholar
[12]Friedland, S. and Peled, U. N. The pressure associated with multidimensional SOFT. In preparation.Google Scholar
[13]Gurvits, L. (2004) Combinatorial and algorithmic aspects of hyperbolic polynomials. arXIv: math.CO/0404474.Google Scholar
[14]Gurvits, L. (2006) Hyperbolic polynomials approach to van der Waerden/Schrijver-Valiant like conjectures: Sharper bounds, simpler proofs and algorithmic applications. In STOC'06: Proc. 38th Annual ACM Symposium on Theory of Computing, ACM, New York, pp. 417–426. arXIv: math.CO/0510452 (2005).CrossRefGoogle Scholar
[15]Hammersley, J. M. (1966) Existence theorems and Monte Carlo methods for the monomer-dimer problem. In Research Papers in Statistics: Festschrift for J. Neyman (David, F. N., ed.), Wiley, London, pp. 125–146.Google Scholar
[16]Heilmann, O. J. and Lieb, E. H. (1972) Theory of monomer-dimer systems. Comm. Math. Phys. 25 190–232.CrossRefGoogle Scholar
[17]Lovász, L. and Plummer, M. D. (1986) Matching Theory, Vol. 121 of North-Holland Mathematical Studies, North-Holland, Amsterdam.Google Scholar
[18]Schrijver, A. (1998) Counting 1-factors in regular bipartite graphs. J. Combin. Theory Ser. B 72 122–135.CrossRefGoogle Scholar
[19]Schrijver, A. and Valiant, W. G. (1980) On lower bounds for permanents. Indagationes Mathematicae 42 425–427.CrossRefGoogle Scholar
[20]Sinkhorn, R. (1964) A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist. 35 876–879.CrossRefGoogle Scholar
[21]Tverberg, H. (1963) On the permanent of bistochastic matrix. Math. Scand. 12 25–35.CrossRefGoogle Scholar
[22]van der Waerden, B. L. (1926) Aufgabe 45. Jahresber. Deutsch. Math.-Verein. 35 117.Google Scholar
[23]Wanless, I. M. (2003) A lower bound on the maximum permanent in Λkn. Linear Algebra Appl. 373 153–167.CrossRefGoogle Scholar