{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T01:59:27Z","timestamp":1769219967676,"version":"3.49.0"},"reference-count":31,"publisher":"Association for Computing Machinery (ACM)","issue":"5","license":[{"start":{"date-parts":[[2016,12,6]],"date-time":"2016-12-06T00:00:00Z","timestamp":1480982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"Cryptography Lend\u00fclet project of the Hungarian Academy of Sciences and the National Research"},{"DOI":"10.13039\/501100011019","name":"NKFIH","doi-asserted-by":"crossref","award":["K-116769 and SNN-117879"],"award-info":[{"award-number":["K-116769 and SNN-117879"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"crossref"}]},{"name":"DFG"},{"name":"Development and Innovation Office"},{"name":"project SZ 261\/1-1"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2016,12,20]]},"abstract":"<jats:p>\n            The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our article is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the\n            <jats:italic>k<\/jats:italic>\n            -SAT problem, where the Local Lemma does give essentially optimal answers.\n          <\/jats:p>\n          <jats:p>\n            As our main contribution, we construct unsatisfiable\n            <jats:italic>k<\/jats:italic>\n            -CNF formulas where every clause has\n            <jats:italic>k<\/jats:italic>\n            distinct literals and every variable appears in at most 2\/e +\n            <jats:italic>o<\/jats:italic>\n            (1))\n            <jats:sup>\n              2\n              <jats:sup>k<\/jats:sup>\n              \/\n              <jats:sub>k<\/jats:sub>\n            <\/jats:sup>\n            clauses. The Lopsided Local Lemma, applied with an assignment of random values according to counterintuitive probabilities, shows that this is asymptotically best possible. The determination of this extremal function is particularly important, as it represents the value where the corresponding\n            <jats:italic>k<\/jats:italic>\n            -SAT problem exhibits a complexity hardness jump: From having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause.\n          <\/jats:p>\n          <jats:p>The construction of our unsatisfiable CNF formulas is based on the binary tree approach of Gebauer [2012], and thus the constructed formulas are in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables. The main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem. This leads us to a differential equation, the solution of which we are able to estimate. The asymptotically optimal binary trees are then obtained through a discretization of this solution.<\/jats:p>\n          <jats:p>\n            The importance of the binary trees constructed is also underlined by their appearance in many other scenarios. In particular, they give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [2004] and a search problem allowing a limited number of consecutive lies. As yet another consequence, we slightly improve the best-known bounds on the maximum degree and maximum edge-degree of a\n            <jats:italic>k<\/jats:italic>\n            -uniform Maker\u2019s win hypergraph in the Neighborhood Conjecture of Beck.\n          <\/jats:p>","DOI":"10.1145\/2975386","type":"journal-article","created":{"date-parts":[[2016,12,6]],"date-time":"2016-12-06T16:03:07Z","timestamp":1481040187000},"page":"1-32","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":10,"title":["The Local Lemma Is Asymptotically Tight for SAT"],"prefix":"10.1145","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7616-3821","authenticated-orcid":false,"given":"Heidi","family":"Gebauer","sequence":"first","affiliation":[{"name":"Zurich University of Applied Sciences"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tibor","family":"Szab\u00f3","sequence":"additional","affiliation":[{"name":"Freie Universit\u00e4t Berlin, Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G\u00e1bor","family":"Tardos","sequence":"additional","affiliation":[{"name":"R\u00e9nyi Institute, Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2016,12,6]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(86)90060-9"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735202"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.5555\/646229.681551"},{"key":"e_1_2_1_4_1","volume-title":"Scott","author":"Berman Piotr","year":"2003","unstructured":"Piotr Berman , Marek Karpinski , and Alex D . Scott . 2003 . Approximation hardness and satisfiability of bounded occurrence instances of SAT. Electronic Colloquium on Computational Complexity (ECCC) 10, 022 (2003). Retrieved from http:\/\/eccc.hpi-web.de\/eccc-reports\/2003\/TR03-022\/index.html. Piotr Berman, Marek Karpinski, and Alex D. Scott. 2003. Approximation hardness and satisfiability of bounded occurrence instances of SAT. Electronic Colloquium on Computational Complexity (ECCC) 10, 022 (2003). Retrieved from http:\/\/eccc.hpi-web.de\/eccc-reports\/2003\/TR03-022\/index.html."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/800157.805047"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018924526592"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2002.10.001"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(90)90020-D"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(73)90005-8"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(91)90040-4"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/285055.285059"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-012-2679-y"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03456-5_3"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1963-0143712-1"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.02.004"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480104445745"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(02)00405-5"},{"key":"e_1_2_1_18_1","volume-title":"Two constructions relating to conjectures of beck on positional games. CoRR abs\/1212.3345","author":"Knox Fiachra","year":"2012","unstructured":"Fiachra Knox . 2012. Two constructions relating to conjectures of beck on positional games. CoRR abs\/1212.3345 ( 2012 ). http:\/\/arxiv.org\/abs\/1212.3345. Fiachra Knox. 2012. Two constructions relating to conjectures of beck on positional games. CoRR abs\/1212.3345 (2012). http:\/\/arxiv.org\/abs\/1212.3345."},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1137\/0222015"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.5555\/792765.793432"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/1667053.1667060"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(91)90023-X"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00303-6"},{"key":"e_1_2_1_24_1","unstructured":"polymath. 2012. (2012). http:\/\/polymathprojects.org\/2012\/07\/12\/minipolymath4-project-imo-2012-q3\/.  polymath. 2012. (2012). http:\/\/polymathprojects.org\/2012\/07\/12\/minipolymath4-project-imo-2012-q3\/."},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(200001)16:1%3C4::AID-RSA2%3E3.0.CO;2-2"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(00)00036-0"},{"key":"e_1_2_1_27_1","volume-title":"Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS","author":"Scheder Dominik","year":"2010","unstructured":"Dominik Scheder . 2010 . Unsatisfiable linear CNF formulas are large and complex . In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010). 621--632. Dominik Scheder. 2010. Unsatisfiable linear CNF formulas are large and complex. In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010). 621--632."},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579368"},{"key":"e_1_2_1_29_1","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(94)90181-3"},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(02)00411-0"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(84)90081-7"}],"container-title":["Journal of the ACM"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2975386","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2975386","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T03:50:18Z","timestamp":1750218618000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2975386"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,6]]},"references-count":31,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2016,12,20]]}},"alternative-id":["10.1145\/2975386"],"URL":"https:\/\/doi.org\/10.1145\/2975386","relation":{},"ISSN":["0004-5411","1557-735X"],"issn-type":[{"value":"0004-5411","type":"print"},{"value":"1557-735X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,6]]},"assertion":[{"value":"2014-11-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-07-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-12-06","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}