Hostname: page-component-5db58dd55d-jnbmb Total loading time: 0 Render date: 2026-05-30T13:19:36.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing the Partition Function for Perfect Matchings in a Hypergraph

Published online by Cambridge University Press:  17 October 2011

ALEXANDER BARVINOK  and
ALEX SAMORODNITSKY
ALEXANDER BARVINOK
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA (e-mail: barvinok@umich.edu)
ALEX SAMORODNITSKY
Affiliation:
Department of Computer Science, Hebrew University of Jerusalem, Givat Ram Campus, 91904, Israel (e-mail: salex@cs.huji.ac.il)
Get access Rights & Permissions [Opens in a new window]

Abstract

Given non-negative weights wS on the k-subsets S of a km-element set V, we consider the sum of the products wS1 ⋅⋅⋅ wSm over all partitions V = S1 ∪ ⋅⋅⋅ ∪Sm into pairwise disjoint k-subsets Si. When the weights wS are positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman–Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 815 - 835
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alon, N. and Friedland, S. (2008) The maximum number of perfect matchings in graphs with a given degree sequence. Electron. J. Combin. 15 #13.CrossRefGoogle Scholar
[2]Barvinok, A. (2011) A bound for the number of vertices of a polytope with applications. arXiv:1108.2871.Google Scholar
[3]Bregman, L. M. (1973) Certain properties of nonnegative matrices and their permanents (in Russian). Dokl. Akad. Nauk SSSR 211 27–30.Google Scholar
[4]Colbourn, C., Hoffman, D. G., Phelps, K. T., Rödl, V. and Winkler, P. M. (1991) The number of t-wise balanced designs. Combinatorica 11 207–218.CrossRefGoogle Scholar
[5]Costello, K. P. and Vu, V. (2009) Concentration of random determinants and permanent estimators. SIAM J. Discrete Math. 23 1356–1371.CrossRefGoogle Scholar
[6]Dow, S. J. and Gibson, P. M. (1987) An upper bound for the permanent of a 3-dimensional (0,1)-matrix. Proc. Amer. Math. Soc. 99 29–34.Google Scholar
[7]Dow, S. J. and Gibson, P. M. (1987) Permanents of d-dimensional matrices. Linear Algebra Appl. 90 133–145.CrossRefGoogle Scholar
[8]Egorychev, G. P. (1981) The solution of van der Waerden's problem for permanents. Adv. Math. 42 299–305.CrossRefGoogle Scholar
[9]Esperet, L., Kardoš, F., King, A. D., Král', D. and Norine, S. (2011) Exponentially many perfect matchings in cubic graphs. Advances in Mathematics 227 1646–1664.CrossRefGoogle Scholar
[10]Falikman, D. I. (1981) Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix (in Russian). Mat. Zametki 29 931–938.Google Scholar
[11]Friedland, S. (2011) Positive diagonal scaling of a nonnegative tensor to one with prescribed slice sums. Linear Algebra Appl. 434 1615–1619.CrossRefGoogle Scholar
[12]Friedland, S. (2011) Analogs of the van der Waerden and Tverberg conjectures for haffnians. arXiv:1102.2542.Google Scholar
[13]Friedland, S., Rider, B. and Zeitouni, O. (2004) Concentration of permanent estimators for certain large matrices. Ann. Appl. Probab. 14 1559–1576.CrossRefGoogle Scholar
[14]Gurvits, L. (2008) Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: One theorem for all. With a corrigendum. Electron. J. Combin. 15 #66.CrossRefGoogle Scholar
[15]Jerrum, M., Sinclair, A. and Vigoda, E. (2004) A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. Assoc. Comput. Mach. 51 671–697.CrossRefGoogle Scholar
[16]Karp, R. M. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations: Proc. Sympos. IBM Thomas J. Watson Res. Center, 1972, Plenum, pp. 85–103.CrossRefGoogle Scholar
[17]Linial, N., Samorodnitsky, A. and Wigderson, A. (2000) A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. Combinatorica 20 545–568.CrossRefGoogle Scholar
[18]Lovász, L. and Plummer, M. D. (2009) Matching Theory, AMS Chelsea Publishing.Google Scholar
[19]Minc, H. (1978) Permanents, Vol. 6 of Encyclopedia of Mathematics and its Applications, Addison-Wesley.Google Scholar
[20]Nesterov, Y. and Nemirovskii, A. (1994) Interior-Point Polynomial Algorithms in Convex Programming, Vol. 13 of SIAM Studies in Applied Mathematics, SIAM.CrossRefGoogle Scholar
[21]Schrijver, A. (1978) A short proof of Minc's conjecture. J. Combin. Theory Ser. A 25 80–83.CrossRefGoogle Scholar
[22]Soules, G. W. (2003) New permanental upper bounds for nonnegative matrices. Linear Multilinear Algebra 51 319–337.CrossRefGoogle Scholar
[23]Valiant, L. G. (1979) The complexity of computing the permanent. Theoret. Comput. Sci. 8 189–201.CrossRefGoogle Scholar
[24]Vu, V. H. (2000) New bounds on nearly perfect matchings in hypergraphs: Higher codegrees do help. Random Struct. Alg. 17 29–63.3.0.CO;2-W>CrossRefGoogle Scholar