{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:37:28Z","timestamp":1787323048676,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2011,1,1]]},"abstract":"<jats:p>We consider the #P-complete problem of counting the number of independent sets in a given graph. Our interest is in understanding the effectiveness of the popular belief propagation (BP) heuristic. BP is a simple iterative algorithm that is known to have at least one fixed point, where each fixed point corresponds to a stationary point of the Bethe free energy (introduced by Yedidia, Freeman, and Weiss [IEEE Trans. Inform. Theory, 51 (2004), pp.\u00a02282\u20132312] in recognition of Bethe\u2019s earlier work in 1935). The evaluation of the Bethe free energy at such a stationary point (or BP fixed point) leads to the Bethe approximation for the number of independent sets of the given graph. BP is not known to converge in general, nor is an efficient, convergent procedure for finding stationary points of the Bethe free energy known. Furthermore, the effectiveness of the Bethe approximation is not well understood. As the first result of this paper we propose a BP-like algorithm that always converges to a stationary point of the Bethe free energy for any graph for the independent set problem. This procedure finds an [Formula: see text]-approximate stationary point in [Formula: see text] iterations for a graph of [Formula: see text] nodes with max-degree [Formula: see text]. We study the quality of the resulting Bethe approximation using the recently developed \u201cloop series\u201d framework of Chertkov and Chernyak [J. Stat. Mech. Theory Exp., 6 (2006), P06009]. As this characterization is applicable only for exact stationary points of the Bethe free energy, we provide a slightly modified characterization that holds for [Formula: see text]-approximate stationary points. We establish that for any graph on [Formula: see text] nodes with max-degree [Formula: see text] and girth larger than [Formula: see text], the multiplicative error between the number of independent sets and the Bethe approximation decays as [Formula: see text] for some [Formula: see text]. This provides a deterministic counting algorithm that leads to strictly different results compared to a recent result of Weitz [in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2006, pp.\u00a0140\u2013149]. Finally, as a consequence of our analysis we prove that the Bethe approximation is exceedingly good for a random 3-regular graph conditioned on the shortest cycle cover conjecture of Alon and Tarsi [SIAM J. Algebr. Discrete Methods, 6 (1985), pp.\u00a0345\u2013350] being true.<\/jats:p>","DOI":"10.1137\/090767145","type":"journal-article","created":{"date-parts":[[2011,7,4]],"date-time":"2011-07-04T19:46:26Z","timestamp":1309808786000},"page":"1012-1034","source":"Crossref","is-referenced-by-count":14,"title":["Counting Independent Sets Using the Bethe Approximation"],"prefix":"10.1137","volume":"25","author":[{"given":"Venkat","family":"Chandrasekaran","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Misha","family":"Chertkov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Gamarnik","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Devavrat","family":"Shah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinwoo","family":"Shin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,7,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0606035"},{"key":"R2","unstructured":"A. Bandyopadhyay and D. Gamarnik, Counting without sampling. New algorithms for enumeration problems using statistical physics, in  Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2006, pp.\u00a0890\u2013899."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(83)90056-4"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2006\/06\/P06009"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539701383844"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1145\/102782.102783"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1999.1071"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199908)15:1<43::AID-RSA3>3.0.CO;2-S"},{"key":"R9","unstructured":"H.O. Georgii,\n                      Gibbs Measures and Phase Transitions\n                      , Walter de Gruyter, Berlin,  1988."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1111\/j.2517-6161.1985.tb01367.x"},{"key":"R11","unstructured":"N. Madras and G. Slade,\n                      The Self-Avoiding Walk. Probability and Its Applications\n                      , Birkh\u00e4user Boston, Boston,  1993."},{"key":"R12","unstructured":"F. Martinelli, A. Sinclair and D. Weitz, Fast mixing for independent sets,  colorings and other models on trees, in\n                      Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms\n                      , ACM, New York, SIAM, Philadelphia,  2004, pp.\u00a0449\u2013458."},{"key":"R13","unstructured":"J. Pearl,\n                      Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference\n                      , Morgan Kaufmann, San Mateo, CA,  1988."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0890-5401(89)90067-9"},{"key":"R15","unstructured":"E. B. Sudderth, M. J. Wainwright and A. S. Willsky, Loop series and Bethe variational bounds in attractive graphical models, in Advances in Neural Information Processing Systems  20, MIT Press, Cambridge, MA, 2008, pp.\u00a01425\u20131432."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.37236\/1552"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1561\/2200000001"},{"key":"R18","unstructured":"D. Weitz, Counting independent sets up to the tree threshold, in  Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2006, pp.\u00a0140\u2013149."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/S0095-8956(81)80022-6"},{"key":"R20","unstructured":"A. B. Yedidia,  Counting Independent Sets and Kernels of Regular Graphs, manuscript, 2009;"},{"key":"R20","unstructured":"available online from http:\/\/arxiv.org\/abs\/0910.4664."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.850085"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090767145","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:39:12Z","timestamp":1787319552000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090767145"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,1]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,1,1]]}},"alternative-id":["10.1137\/090767145"],"URL":"https:\/\/doi.org\/10.1137\/090767145","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,1]]}}}