The Annals of Probability

Hafnians, perfect matchings and Gaussian matrices

Mark Rudelson, Alex Samorodnitsky, and Ofer Zeitouni

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We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with nonnegative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude that Barvinok’s estimator gives a polynomial-time algorithm for the approximate (up to subexponential errors) evaluation of the number of perfect matchings.

Article information

Ann. Probab., Volume 44, Number 4 (2016), 2858-2888.

Received: September 2014
Revised: June 2015
First available in Project Euclid: 2 August 2016

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices
Secondary: 05C70: Factorization, matching, partitioning, covering and packing

Hafnian perfect matching random Gaussian matrices


Rudelson, Mark; Samorodnitsky, Alex; Zeitouni, Ofer. Hafnians, perfect matchings and Gaussian matrices. Ann. Probab. 44 (2016), no. 4, 2858--2888. doi:10.1214/15-AOP1036.

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  • [1] Ajanki, O., Erdős, L. and Krüger, T. (2014). Local semicircle law with imprimitive variance matrix. Electron. Commun. Probab. 19 no. 33, 9.
  • [2] Barvinok, A. (1999). Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor. Random Structures Algorithms 14 29–61.
  • [3] Barvinok, A. and Samorodnitsky, A. (2007). Random weighting, asymptotic counting, and inverse isoperimetry. Israel J. Math. 158 159–191.
  • [4] Barvinok, A. and Samorodnitsky, A. (2011). Computing the partition function for perfect matchings in a hypergraph. Combin. Probab. Comput. 20 815–835.
  • [5] Bayati, M., Gamarnik, D., Katz, D., Nair, C. and Tetali, P. (2007). Simple deterministic approximation algorithms for counting matchings. In STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing 122–127. ACM, New York.
  • [6] Brègman, L. M. (1973). Some properties of nonnegative matrices and their permanents. Soviet Math. Dokl. 211 945–949.
  • [7] Erdős, L., Knowles, A., Yau, H.-T. and Yin, J. (2013). The local semicircle law for a general class of random matrices. Electron. J. Probab. 18 no. 59, 58.
  • [8] Friedland, S., Rider, B. and Zeitouni, O. (2004). Concentration of permanent estimators for certain large matrices. Ann. Appl. Probab. 14 1559–1576.
  • [9] Godsil, C. D. and Gutman, I. (1981). On the matching polynomial of a graph. In Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978). (L. Lóvasz and V. T. Sós, eds.) 241–249. North-Holland, Amsterdam.
  • [10] Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 5 119–136.
  • [11] Jerrum, M. and Sinclair, A. (1989). Approximating the permanent. SIAM J. Comput. 18 1149–1178.
  • [12] Jerrum, M., Sinclair, A. and Vigoda, E. (2004). A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51 671–697.
  • [13] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [14] Linial, N., Samorodnitsky, A. and Wigderson, A. (2000). A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. Combinatorica 20 545–568.
  • [15] Minc, H. (1978). Permanents. Encyclopedia of Mathematics and Its Applications 6. Addison-Wesley, Reading, MA.
  • [16] Rudelson, M. and Vershynin, R. (2008). The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218 600–633.
  • [17] Rudelson, M. and Vershynin, R. (2009). Smallest singular value of a random rectangular matrix. Comm. Pure Appl. Math. 62 1707–1739.
  • [18] Rudelson, M. and Zeitouni, O. (2014). Singular values of Gaussian matrices and permanent estimators. Random Structures Algorithms. To appear. Available at arXiv:1301.6268.
  • [19] Valiant, L. G. (1979). The complexity of computing the permanent. Theoret. Comput. Sci. 8 189–201.