Hostname: page-component-546b4f848f-q5mmw Total loading time: 0 Render date: 2023-06-03T13:48:44.627Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Counting matchings via capacity-preserving operators

Published online by Cambridge University Press:  30 April 2021

Leonid Gurvits
City College of New York, New York, NY, USA
Jonathan Leake*
KTH, Stockholm, Sweden
*Corresponding author. Email:
Rights & Permissions[Opens in a new window]


HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2021. Published by Cambridge University Press


Anari, N. and Gharan, S. O. (2017) A generalization of permanent inequalities and applications in counting and optimization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, pp. 384396.CrossRefGoogle Scholar
Anari, N., Gharan, S. O. and Vinzant, C. (2018) Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, pp. 3546.CrossRefGoogle Scholar
Adiprasito, K., Huh, J. and Katz, E. (2018) Hodge theory for combinatorial geometries. Ann. Math. 188(2) 381452.CrossRefGoogle Scholar
Bapat, R. B. (1989) Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl. 126 107124.CrossRefGoogle Scholar
Borcea, J. and Brändén, P. (2009) The Lee-Yang and Pólya-Schur programs I: Linear operators preserving stability. Inventiones Mathematicae 177(3) 541569.CrossRefGoogle Scholar
Borcea, J. and Brändén, P. (2009) The Lee-Yang and Pólya-Schur programs II: Theory of stable polynomials and applications. Commun. Pure Appl. Math. 62(12) 15951631.CrossRefGoogle Scholar
Bürgisser, P., Garg, A., Oliveira, R., Walter, M. and Wigderson, A. (2017) Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory. arXiv preprint arXiv:1711.08039.Google Scholar
Brändén, P. and Huh, J. (2019) Lorentzian polynomials. arXiv preprint arXiv:1902.03719.Google Scholar
Brändén, P. (2007) Polynomials with the half-plane property and matroid theory. Adv. Math. 216(1) 302320.CrossRefGoogle Scholar
Craven, T. and Csordas, G. (1989) Jensen polynomials and the Turán and Laguerre inequalities. Pac. J. Math. 136(2) 241260.CrossRefGoogle Scholar
Choe, Y.-B., Oxley, J. G., Sokal, A. D. and Wagner, D. G. (2004) Homogeneous multivariate polynomials with the half-plane property. Adv. Appl. Math. 32(1–2) 88187.CrossRefGoogle Scholar
Csikvári, P. (2014) Lower matching conjecture, and a new proof of Schrijver’s and Gurvits’s theorems. arXiv preprint arXiv:1406.0766.Google Scholar
Egorychev, G. P. (1981) Proof of the van der Waerden conjecture for permanents. Siberian Math. J. 22(6) 854859.CrossRefGoogle Scholar
Falikman, D. I. (1981) Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix. Math. Notes Acad. Sci. USSR 29(6) 475479.Google Scholar
Friedland, S. and Gurvits, L. (2008) Lower bounds for partial matchings in regular bipartite graphs and applications to the monomer–dimer entropy. Comb. Prob. Comput. 17(3) 347361.CrossRefGoogle Scholar
Friedland, S., Krop, E. and Markström, K. (2008) On the number of matchings in regular graphs. Electr. J. Comb. 15(1) 110.CrossRefGoogle Scholar
Garg, A., Gurvits, L., Oliveira, R. and Wigderson, A. (2015) Operator scaling: Theory and applications. arXiv preprint arXiv:1511.03730.Google Scholar
Garg, A., Gurvits, L., Oliveira, R. and Wigderson, A. (2016) A deterministic polynomial time algorithm for non-commutative rational identity testing. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, pp. 109117.CrossRefGoogle Scholar
Gurvits, L. (2006) Hyperbolic polynomials approach to Van der Waerden/Schrijver-Valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, ACM, pp. 417426.CrossRefGoogle Scholar
Gurvits, L. (2008) Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all. Electr. J. Comb. 15(1) 66.CrossRefGoogle Scholar
Gurvits, L. (2009) On multivariate Newton-like inequalities. In Advances in Combinatorial Mathematics, Springer, pp. 6178.CrossRefGoogle Scholar
Gurvits, L. (2011) Unleashing the power of Schrijver’s permanental inequality with the help of the Bethe approximation. arXiv preprint arXiv:1106.2844.Google Scholar
Heilmann, O. J. and Lieb, E. H. (1972) Theory of monomer-dimer systems. In Statistical Mechanics, Springer, Berlin, Heidelberg, pp. 4587.CrossRefGoogle Scholar
Huh, J., Schröter, B. and Wang, B. (2018) Correlation bounds for fields and matroids. arXiv preprint arXiv:1806.02675.Google Scholar
Leake, J. (2017) A representation theoretic explanation of the Borcea-Brändén characterization. arXiv preprint arXiv:1706.06168.Google Scholar
Marcus, A., Spielman, D. and Srivastava, N. (2015) Finite free convolutions of polynomials. arXiv preprint arXiv:1504.00350.Google Scholar
Marcus, A., Spielman, D. and Srivastava, N. (2015) Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem. Ann. Math. 182 327350.CrossRefGoogle Scholar
Renegar, J. (2006) Hyperbolic programs, and their derivative relaxations. Found. Comput. Math. 6(1) 5979.CrossRefGoogle Scholar
Schrijver, A. (1998) Counting 1-factors in regular bipartite graphs. J. Comb. Theory, Ser. B 72(1) 122135.CrossRefGoogle Scholar
Straszak, D. and Vishnoi, N. K. (2017) Real stable polynomials and matroids: Optimization and counting. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, pp. 370383.CrossRefGoogle Scholar
Walsh, J. L. (1922) On the location of the roots of certain types of polynomials. Trans. Am. Math. Soc. 24(3) 163180.CrossRefGoogle Scholar
Zackrisson, S. (2017) Coefficients and zeros of mixed characteristic polynomials.Google Scholar