{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T06:58:24Z","timestamp":1791269904070,"version":"4.1.0"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterized by a finite set $\\mathcal{F}$ of nonnegative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e., the sum of the weights of all configurations, is $\\text{{\\sf FP}}^{\\text{{\\sf#P}}}$-complete unless either (1) every function in $\\mathcal{F}$ is of \u201cproduct type,\u201d or (2) every function in $\\mathcal{F}$ is \u201cpure affine.\u201d In the remaining cases, computing the partition function is in P.<\/jats:p>","DOI":"10.1137\/070690201","type":"journal-article","created":{"date-parts":[[2009,1,15]],"date-time":"2009-01-15T10:43:15Z","timestamp":1232016195000},"page":"1970-1986","source":"Crossref","is-referenced-by-count":66,"title":["The Complexity of Weighted Boolean #CSP"],"prefix":"10.1137","volume":"38","author":[{"given":"Martin","family":"Dyer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leslie Ann","family":"Goldberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Jerrum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,1,14]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1899"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"A. Bulatov and V. Dalmau,\n                      Towards a dichotomy theorem for the counting constraint satisfaction problem\n                      , in Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2003, pp. 562\u2013573.","DOI":"10.1109\/SFCS.2003.1238229"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2006.04.002"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.09.011"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1987.12000742"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1006\/inco.1996.0016"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"N. Creignou, S. Khanna, and M. Sudan,\n                      Complexity Classifications of Boolean Constraint Satisfaction Problems\n                      , SIAM Monogr. Discrete Math. Appl. 7, SIAM, Philadelphia, 2001.","DOI":"10.1137\/1.9780898718546"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200010\/12)17:3\/4<260::AID-RSA5>3.0.CO;2-W"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"M. Dyer, L. A. Goldberg, and M. Paterson,\n                      On counting homomorphisms to directed acyclic graphs\n                      , in Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, Lecture Notes in Comput. Sci. 4051, Springer, New York, 2006, pp. 38\u201349.","DOI":"10.1007\/11786986_5"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539794266766"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2008.04.003"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/PL00001601"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"P. Hell and J. Ne\u0161et\u0159il,\n                      Graphs and Homomorphisms\n                      , Oxford University Press, London, 2004.","DOI":"10.1093\/acprof:oso\/9780198528173.001.0001"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02280291"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1145\/321864.321877"},{"key":"R16","unstructured":"C. Papadimitriou,\n                      Computational Complexity\n                      , Addison\u2013Wesley, Reading, MA, 1994."},{"key":"R17","unstructured":"F. Rossi, P. van Beek, and T. Walsh, eds.\n                      Handbook of Constraint Programming\n                      , Elsevier, New York, 2006."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"T. Schaefer,\n                      The complexity of satisfiability problems\n                      , in Proceedings of the 10th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 1978, pp. 216\u2013226.","DOI":"10.1145\/800133.804350"},{"key":"R19","unstructured":"A. Scott and G. Sorkin,\n                      Polynomial Constraint Satisfaction: A Framework for Counting and Sampling CSPs and Other Problems\n                      , online at http:\/\/arxiv.org\/abs\/cs\/0604079."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1145\/322217.322225"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0208032"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"D. Welsh,\n                      Complexity: Knots, Colourings and Counting\n                      , LMS Lecture Note Ser. 186,Cambridge University Press, Cambridge, UK, 1993.","DOI":"10.1017\/CBO9780511752506"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070690201","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:22:42Z","timestamp":1787336562000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070690201"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":22,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/070690201"],"URL":"https:\/\/doi.org\/10.1137\/070690201","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}