{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:14:10Z","timestamp":1750306450024,"version":"3.41.0"},"reference-count":14,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2015,5,11]],"date-time":"2015-05-11T00:00:00Z","timestamp":1431302400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-0747250"],"award-info":[{"award-number":["CCF-0747250"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001742","name":"United States-Israel Binational Science Foundation","doi-asserted-by":"crossref","award":["2008477"],"award-info":[{"award-number":["2008477"]}],"id":[{"id":"10.13039\/501100001742","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2015,5,11]]},"abstract":"<jats:p>\n            In 1997, H\u00e5stad showed NP-hardness of (1 \u2212 \u03f5, 1\/\n            <jats:italic>q<\/jats:italic>\n            + \u03b4)-approximating Max-3Lin(Z\n            <jats:sub>q<\/jats:sub>\n            ); however, it was not until 2007 that Guruswami and Raghavendra were able to show NP-hardness of (1 \u2212 \u03f5, \u03b4)-approximating Max-3Lin(Z). In 2004, Khot--Kindler--Mossel--O\u2019Donnell showed UG-hardness of (1 \u2212 \u03f5, \u03b4)-approximating Max-2Lin(Z\n            <jats:sub>q<\/jats:sub>\n            ) for\n            <jats:italic>q<\/jats:italic>\n            =\n            <jats:italic>q<\/jats:italic>\n            (\u03f5, \u03b4) a sufficiently large constant; however, achieving the same hardness for Max-2Lin(Z) was given as an open problem in Raghavendra\u2019s 2009 thesis. In this work, we show that fairly simple modifications to the proofs of the Max-3Lin(Z\n            <jats:sub>q<\/jats:sub>\n            ) and Max-2Lin(Z\n            <jats:sub>q<\/jats:sub>\n            ) results yield optimal hardness results over Z. In fact, we show a kind of \u201cbicriteria\u201d hardness: Even when there is a (1 \u2212 \u03f5)-good solution over Z, it is hard for an algorithm to find a \u03b4-good solution over Z, R, or Z\n            <jats:sub>m<\/jats:sub>\n            for any\n            <jats:italic>m<\/jats:italic>\n            \u2a7e\n            <jats:italic>q<\/jats:italic>\n            (\u03f5, \u03b4) of the algorithm\u2019s choosing.\n          <\/jats:p>","DOI":"10.1145\/2751322","type":"journal-article","created":{"date-parts":[[2015,5,12]],"date-time":"2015-05-12T12:18:00Z","timestamp":1431433080000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Hardness of Max-2Lin and Max-3Lin over Integers, Reals, and Large Cyclic Groups"],"prefix":"10.1145","volume":"7","author":[{"given":"Ryan","family":"O\u2019Donnell","sequence":"first","affiliation":[{"name":"Carnegie Mellon University, Pittsburgh, PA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yi","family":"Wu","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University, West Lafayette, IN"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Zhou","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University, Pittsburgh, PA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2015,5,11]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1993.366815"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/278298.278306"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/273865.273901"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2006.51"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2006.33"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/1250790.1250819"},{"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":"10.1145\/509907.510017"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539705447372"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/110846415"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.2005.53"},{"key":"e_1_2_1_13_1","volume-title":"Analysis of Boolean Functions Lecture Notes","author":"O\u2019Donnell Ryan","year":"2007","unstructured":"Ryan O\u2019Donnell . 2007. Analysis of Boolean Functions Lecture Notes , 2007 . Retrieved from http:\/\/www.cs.cmu.edu\/&sim;odonnell\/boolean-analysis\/. Ryan O\u2019Donnell. 2007. Analysis of Boolean Functions Lecture Notes, 2007. Retrieved from http:\/\/www.cs.cmu.edu\/&sim;odonnell\/boolean-analysis\/."},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539795280895"}],"container-title":["ACM Transactions on Computation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2751322","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2751322","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T05:43:06Z","timestamp":1750225386000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2751322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,11]]},"references-count":14,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,5,11]]}},"alternative-id":["10.1145\/2751322"],"URL":"https:\/\/doi.org\/10.1145\/2751322","relation":{},"ISSN":["1942-3454","1942-3462"],"issn-type":[{"type":"print","value":"1942-3454"},{"type":"electronic","value":"1942-3462"}],"subject":[],"published":{"date-parts":[[2015,5,11]]},"assertion":[{"value":"2013-08-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2014-06-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2015-05-11","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}