{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:25:34Z","timestamp":1787340334303,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be \u201cnontrivial.\u201d Here is an informal statement of our result: it is $\\mathcal{NP}$-hard to distinguish whether there is a nontrivial assignment that satisfies $1-\\delta$ fraction of the equations or every nontrivial assignment fails to satisfy a constant fraction of the equations with a \u201cmargin\u201d of $\\Omega(\\sqrt{\\delta})$. We develop linearity and dictatorship testing procedures for functions $f: \\mathbb{R}^n \\mapsto \\mathbb{R}$ over a Gaussian space, which could be of independent interest. Our research is motivated by a possible approach to proving the unique games conjecture.<\/jats:p>","DOI":"10.1137\/110846415","type":"journal-article","created":{"date-parts":[[2013,5,2]],"date-time":"2013-05-02T18:12:15Z","timestamp":1367518335000},"page":"752-791","source":"Crossref","is-referenced-by-count":3,"title":["$\\mathcal{NP}$-Hardness of Approximately Solving Linear Equations over Reals"],"prefix":"10.1137","volume":"42","author":[{"given":"Subhash","family":"Khot","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dana","family":"Moshkovitz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,5,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"S. Arora, B. Barak, and D. Steurer,\n                      Subexponential algorithms for unique games and related problems\n                      , in Proceedings of the 51st IEEE Symposium on Foundations of Computer Science, Las Vegas, 2010.","DOI":"10.1109\/FOCS.2010.59"},{"key":"atypb2","first-page":"21","author":"Arora S.","year":"2008","journal-title":"Victoria, BC"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1145\/278298.278306"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1145\/273865.273901"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1109\/18.556674"},{"key":"atypb6","first-page":"21","author":"Babai L.","year":"1991","journal-title":"New Orleans"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(93)90044-W"},{"key":"atypb8","unstructured":"T. M. Cover and J. Thomas,\n                      Elements of Information Theory\n                      , Wiley, New York, 1991."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1145\/226643.226652"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1145\/1595391.1595393"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1145\/502090.502098"},{"key":"atypb12","first-page":"11","author":"Holmerin J.","year":"2004","journal-title":"Chicago"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2009.v005a008"},{"key":"atypb14","first-page":"767","author":"Khot S.","year":"2002","journal-title":"Montreal"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539705447372"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"S. Khot and D. Moshkovitz,\n                      Hardness of approximately solving linear equations over reals\n                      , Technical report, ECCC Report TR10-053, 2010.","DOI":"10.1145\/1993636.1993692"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1145\/585265.585268"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"D. Pollard,\n                      A User's Guide to Measure Theoretic Probability\n                      , Cambridge University Press, Cambridge, 2002.","DOI":"10.1017\/CBO9780511811555"},{"key":"atypb19","first-page":"245","author":"Raghavendra P.","year":"2008","journal-title":"Victoria, BC"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539795280895"},{"key":"atypb21","first-page":"201","author":"Zwick U.","year":"1998","journal-title":"ACM--SIAM"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/110846415","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:30:33Z","timestamp":1787337033000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/110846415"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["10.1137\/110846415"],"URL":"https:\/\/doi.org\/10.1137\/110846415","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}