{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T08:22:55Z","timestamp":1790324575847,"version":"4.1.0"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>In the constraint satisfaction problem (CSP) corresponding to a constraint language (i.e., a set of relations) $\\Gamma$, the goal is to find an assignment of values to variables so that a given set of constraints specified by relations from $\\Gamma$ is satisfied. The complexity of this problem has received a substantial amount of attention in the past decade. In this paper, we study the fixed-parameter tractability of CSPs parameterized by the size of the solution in the following sense: one of the possible values, say 0, is \u201cfree,\u201d and the number of variables allowed to take other, \u201cexpensive,\u201d values is restricted. A size constraint requires that exactly $k$ variables take nonzero values. We also study a more refined version of this restriction: a global cardinality constraint prescribes how many variables have to be assigned each particular value. We study the parameterized complexity of these types of CSPs where the parameter is the required number $k$ of nonzero variables. As special cases, we can obtain natural and well-studied parameterized problems such as Independent set, Vertex Cover, $d$-Hitting Set, Biclique, etc. In the case of constraint languages closed under substitution of constants, we give a complete characterization of the fixed-parameter tractable cases of CSPs with size constraints, and we show that all the remaining problems are W[1]-hard. For CSPs with cardinality constraints, we obtain a similar classification, but for some of the problems we are only able to show that they are Biclique-hard. The exact parameterized complexity of the Biclique problem is a notorious open problem, although it is believed to be W[1]-hard.<\/jats:p>","DOI":"10.1137\/120882160","type":"journal-article","created":{"date-parts":[[2014,4,15]],"date-time":"2014-04-15T12:06:40Z","timestamp":1397563600000},"page":"573-616","source":"Crossref","is-referenced-by-count":9,"title":["Constraint Satisfaction Parameterized by Solution Size"],"prefix":"10.1137","volume":"43","author":[{"given":"Andrei A.","family":"Bulatov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D\u00e1niel","family":"Marx","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,4,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"L. Barto, M. Kozik, and T. Niven,\n                      Graphs, polymorphisms and the complexity of homomorphism problems\n                      , in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 789-796.","DOI":"10.1145\/1374376.1374488"},{"key":"atypb2","first-page":"112","author":"Bessi\u00e8re C.","year":"2004","journal-title":"MA"},{"key":"atypb3","first-page":"153","author":"Bourdais S.","year":"2003","journal-title":"Heidelberg"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539700376676"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1145\/1120582.1120584"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1145\/1970398.1970400"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.2168\/LMCS-6(4:4)2010"},{"key":"atypb8","first-page":"424","author":"Bulatov A. A.","year":"2011","journal-title":"Heidelberg"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1145\/1810891.1810914"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1145\/1805950.1805954"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"R. G. Downey and M. R. Fellows,\n                      Parameterized Complexity\n                      , Monogr. Comput. Sci., Springer, New York, 1999.","DOI":"10.1007\/978-1-4612-0515-9"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539794266766"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2008.09.065"},{"key":"atypb14","unstructured":"J. Flum and M. Grohe,\n                      Parameterized Complexity Theory\n                      , Springer, Berlin, 2006."},{"key":"atypb15","unstructured":"S. Gaspers and S. Szeider,\n                      Kernels for global constraints\n                      , in Proceedings of the 22nd International Joint Conference on Artificial Intelligence, AAAI Press, Palo Alto, CA, 2011, pp. 540-545."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1145\/1206035.1206036"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1145\/263867.263489"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009890709297"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539799349948"},{"key":"atypb20","first-page":"489","author":"Kratsch S.","year":"2010","journal-title":"Heidelberg"},{"key":"atypb21","first-page":"653","author":"Kratsch S.","year":"2010","journal-title":"Heidelberg"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1145\/2151171.2151182"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"G. Kun and M. Szegedy,\n                      A new line of attack on the dichotomy conjecture\n                      , in Proceedings of the 2009 ACM International Symposium on Theory of Computing, 2009, pp. 725-734.","DOI":"10.1145\/1536414.1536512"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-005-0195-9"},{"key":"atypb25","first-page":"572","author":"R\u00e9gin J.-C.","year":"2004","journal-title":"Heidelberg"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"I. G. Rosenberg,\n                      Multiple-valued hyperstructures\n                      , in Proceedings of the 28th IEEE International Symposium on Multiple-Valued Logic (ISMVL '98), 1998, pp. 326-333.","DOI":"10.1109\/ISMVL.1998.679509"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s10601-009-9079-y"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1145\/800133.804350"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.disopt.2010.07.003"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120882160","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:51:13Z","timestamp":1787338273000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120882160"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1137\/120882160"],"URL":"https:\/\/doi.org\/10.1137\/120882160","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1]]}}}