{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:19:46Z","timestamp":1750306786922,"version":"3.41.0"},"reference-count":21,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2013,11,1]],"date-time":"2013-11-01T00:00:00Z","timestamp":1383264000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000143","name":"Division of Computing and Communication Foundations","doi-asserted-by":"publisher","award":["CCF-1118083, CCF-1320105, DMS-1106999"],"award-info":[{"award-number":["CCF-1118083, CCF-1320105, DMS-1106999"]}],"id":[{"id":"10.13039\/100000143","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["N000141110140"],"award-info":[{"award-number":["N000141110140"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["CCF-1118083, CCF-1320105, DMS-1106999"],"award-info":[{"award-number":["CCF-1118083, CCF-1320105, DMS-1106999"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2013,11]]},"abstract":"<jats:p>The results of Raghavendra [2008] show that assuming Khot\u2019s Unique Games Conjecture [2002], for every constraint satisfaction problem there exists a generic semidefinite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem.<\/jats:p>\n          <jats:p>Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. Khot et al. [2004] established a general approach for determining the exact approximation factor and the corresponding optimal rounding algorithm for any given constraint satisfaction problem. However, this approach crucially relies on results explicitly proving optimal partitions in the Gaussian space. Until recently, Borell\u2019s result [1985] was the only nontrivial Gaussian partition result known.<\/jats:p>\n          <jats:p>In this article we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel [2012]. This Gaussian result allows us to determine the exact Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick\u2019s algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is \u2248 0.796 as conjectured by Zwick [1998].<\/jats:p>\n          <jats:p>\n            We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-k-CSP: Using the well-known fact that jointly normal pairwise independent random variables are fully independent, we show that the UGC hardness of Max-k-CSP is \u2308(\n            <jats:italic>k<\/jats:italic>\n            +1)\/2\u2309 2\n            <jats:sup>k\u22121<\/jats:sup>\n            , improving on results of Austrin and Mossel [2009].\n          <\/jats:p>","DOI":"10.1145\/2505766","type":"journal-article","created":{"date-parts":[[2013,12,10]],"date-time":"2013-12-10T13:28:12Z","timestamp":1386682092000},"page":"1-26","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Explicit Optimal Hardness via Gaussian Stability Results"],"prefix":"10.1145","volume":"5","author":[{"given":"Anindya","family":"De","sequence":"first","affiliation":[{"name":"University of California, Berkeley"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elchanan","family":"Mossel","sequence":"additional","affiliation":[{"name":"University of California, Berkeley"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2013,11]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1250790.1250818"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/CCC.2008.20"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177704254"},{"key":"e_1_2_1_4_1","unstructured":"Benjamini I. Gurel-Gurevich O. and Peled R. 2012. On k-wise independent distributions and Boolean functions. http:\/\/arxiv.org\/abs\/1201.3261.  Benjamini I. Gurel-Gurevich O. and Peled R. 2012. On k-wise independent distributions and Boolean functions. http:\/\/arxiv.org\/abs\/1201.3261."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00532234"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1287\/moor.14.2.317"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/227683.227684"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/502090.502098"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-011-0181-7"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/026\/737400"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/509907.510017"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2007.06.019"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539705447372"},{"key":"e_1_2_1_14_1","doi-asserted-by":"crossref","unstructured":"Lewin M. Livnat D. and Zwick U. 2002. Improved rounding techniques for Max-Di-Cut and Max-2-Sat. In Integer Programming and Combinatorial Optimization 67--82.   Lewin M. Livnat D. and Zwick U. 2002. Improved rounding techniques for Max-Di-Cut and Max-2-Sat. In Integer Programming and Combinatorial Optimization 67--82.","DOI":"10.1007\/3-540-47867-1_6"},{"volume-title":"Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX\u201912)","author":"Makarychev K.","key":"e_1_2_1_15_1","unstructured":"Makarychev , K. and Makarychev , Y . 2012. Approximation algorithm for non-Boolean Max k-CSP . In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX\u201912) . 254--265. Makarychev, K. and Makarychev, Y. 2012. Approximation algorithm for non-Boolean Max k-CSP. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX\u201912). 254--265."},{"key":"e_1_2_1_16_1","unstructured":"Mossel E. 2005. Lecture notes on fourier analysis. http:\/\/www.stat.berkeley.edu\/~mossel\/teach\/206af05\/.  Mossel E. 2005. Lecture notes on fourier analysis. http:\/\/www.stat.berkeley.edu\/~mossel\/teach\/206af05\/."},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-010-0047-x"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2010.171.295"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/1374376.1374414"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/1132516.1132519"},{"key":"e_1_2_1_21_1","volume-title":"Proceedings of the 9th Annual ACM\/SIGACT-SIAM Symposium on Discrete Algorithms (SODA\u201998)","author":"Zwick U.","year":"1998","unstructured":"Zwick , U. 1998 . Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint . In Proceedings of the 9th Annual ACM\/SIGACT-SIAM Symposium on Discrete Algorithms (SODA\u201998) . 201--210. Zwick, U. 1998. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the 9th Annual ACM\/SIGACT-SIAM Symposium on Discrete Algorithms (SODA\u201998). 201--210."}],"container-title":["ACM Transactions on Computation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2505766","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2505766","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T07:34:16Z","timestamp":1750232056000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2505766"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2013,11]]}},"alternative-id":["10.1145\/2505766"],"URL":"https:\/\/doi.org\/10.1145\/2505766","relation":{},"ISSN":["1942-3454","1942-3462"],"issn-type":[{"type":"print","value":"1942-3454"},{"type":"electronic","value":"1942-3462"}],"subject":[],"published":{"date-parts":[[2013,11]]},"assertion":[{"value":"2012-06-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2013-06-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2013-11-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}