Skip to main content Accessibility help
×
Home
Hostname: page-component-77ffc5d9c7-866w8 Total loading time: 0.41 Render date: 2021-04-23T10:36:52.623Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Approximately counting bases of bicircular matroids

Published online by Cambridge University Press:  06 August 2020

Heng Guo
Affiliation:
School of Informatics, University of Edinburgh, Informatics Forum, Edinburgh EH8 9AB, UK
Mark Jerrum
Affiliation:
School ofMathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
Corresponding
E-mail address:

Abstract

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.

Type
Paper
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

The work described here was supported by the EPSRC research grant EP/N004221/1 ‘Algorithms that Count’.

References

Anari, N., Liu, K., Oveis Gharan, S. and Vinzant, C. (2019) Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), pp. 112. ACM.CrossRefGoogle Scholar
Chávez Lomelí, L. and Welsh, D. J. A. (1996) Randomized approximation of the number of bases. In Matroid Theory (Seattle, WA, 1995), Vol. 197 of Contemporary Mathematics, pp. 371376. AMS.Google Scholar
Cohn, H., Pemantle, R. and Propp, J. G. (2002) Generating a random sink-free orientation in quadratic time. Electron. J. Combin. 9 #R10.CrossRefGoogle Scholar
Cryan, M., Guo, H. and Mousa, G. (2019) Modified log-Sobolev inequalities for strongly log-concave distributions. In 60th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2019), pp. 13581370. IEEE.Google Scholar
Eriksson, K. (1996) Strong convergence and a game of numbers. European J. Combin. 17 379390.Google Scholar
Feder, T. and Mihail, M. (1992) Balanced matroids. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC ’92), pp. 2638. ACM.Google Scholar
Giménez, O., de Mier, A. and Noy, M. (2005) On the number of bases of bicircular matroids. Ann. Combin. 9 3545.Google Scholar
Giménez, O. and Noy, M. (2006) On the complexity of computing the Tutte polynomial of bicircular matroids. Combin. Probab. Comput. 15 385395.CrossRefGoogle Scholar
Gorodezky, I. and Pak, I. (2014) Generalized loop-erased random walks and approximate reachability. Random Struct. Algorithms 44 201223.Google Scholar
Guo, H. and He, K. (2018) Tight bounds for popping algorithms. Random Struct. Algorithms. doi: 10.1002/rsa.20928.Google Scholar
Guo, H. and Jerrum, M. (2019) A polynomial-time approximation algorithm for all-terminal network reliability. SIAM J. Comput. 48 964978.CrossRefGoogle Scholar
Guo, H., Jerrum, M. and Liu, J. (2019) Uniform sampling through the Lovász local lemma. J. Assoc. Comput. Mach. 66 18.Google Scholar
Huber, M. (2015) Approximation algorithms for the normalizing constant of Gibbs distributions. Ann. Appl. Probab. 25 974985.CrossRefGoogle Scholar
Jerrum, M. (2003) Counting, Sampling and Integrating: Algorithms and Complexity, Lectures in Mathematics ETH Zürich. Birkhäuser.CrossRefGoogle Scholar
Jerrum, M. (2006) Two remarks concerning balanced matroids. Combinatorica 26 733742.Google Scholar
Jerrum, M., Son, J.-B., Tetali, P. and Vigoda, E. (2004) Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains. Ann. Appl. Probab. 14 17411765.CrossRefGoogle Scholar
Kassel, A. (2015) Learning about critical phenomena from scribbles and sandpiles. In Modélisation Aléatoire et Statistique: Journées MAS 2014, Vol. 51 of ESAIM Proc. Surveys, pp. 6073. EDP Sciences, Les Ulis.CrossRefGoogle Scholar
Kassel, A. and Kenyon, R. (2017) Random curves on surfaces induced from the Laplacian determinant. Ann. Probab. 45 932964.Google Scholar
Kolipaka, K. B. R. and Szegedy, M. (2011) Moser and Tardos meet Lovász. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC 2011), pp. 235244. ACM.Google Scholar
Kolmogorov, V. (2018) A faster approximation algorithm for the Gibbs partition function. In Proceedings of the 31st Conference On Learning Theory (COLT), Vol. 75 of Proceedings of Machine Learning Research, pp. 228249. PMLR.Google Scholar
Matthews, L. R. (1977) Bicircular matroids. Quart. J. Math. Oxford Ser. (2) 28 213227.CrossRefGoogle Scholar
Maurer, S. B. (1976) Matrix generalizations of some theorems on trees, cycles and cocycles in graphs. SIAM J. Appl. Math. 30 143148.CrossRefGoogle Scholar
Motwani, R. and Raghavan, P. (1995) Randomized Algorithms. Cambridge University Press.CrossRefGoogle Scholar
Piff, M. J. and Welsh, D. J. A. (1971) The number of combinatorial geometries. Bull. London Math. Soc. 3 5556.Google Scholar
Propp, J. G. and Wilson, D. B. (1998) How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms 27 170217.Google Scholar
Snook, M. (2012) Counting bases of representable matroids. Electron. J. Combin. 19 P41.CrossRefGoogle Scholar
Štefankovič, D., Vempala, S. and Vigoda, E. (2009) Adaptive simulated annealing: a near-optimal connection between sampling and counting. J. Assoc. Comput. Mach. 56 18.CrossRefGoogle Scholar
Wilson, D. B. (1996) Generating random spanning trees more quickly than the cover time. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC 1996), pp. 296303. ACM.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 34 *
View data table for this chart

* Views captured on Cambridge Core between 06th August 2020 - 23rd April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Approximately counting bases of bicircular matroids
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Approximately counting bases of bicircular matroids
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Approximately counting bases of bicircular matroids
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *