{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T14:26:27Z","timestamp":1775571987431,"version":"3.50.1"},"reference-count":44,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2015,3,2]],"date-time":"2015-03-02T00:00:00Z","timestamp":1425254400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2015,3,2]]},"abstract":"<jats:p>\n            In this article, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into \u2113\n            <jats:sub>1<\/jats:sub>\n            with constant distortion. We show that for an arbitrarily small constant \u03b4 &gt; 0, for all large enough\n            <jats:italic>n<\/jats:italic>\n            , there is an\n            <jats:italic>n<\/jats:italic>\n            -point negative type metric which requires distortion at least (log log\n            <jats:italic>n<\/jats:italic>\n            )\n            <jats:sup>1\/6-\u03b4<\/jats:sup>\n            to embed into \u2113\n            <jats:sub>1<\/jats:sub>\n            . Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC), establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (nonuniform) S\n            <jats:sc>PARSEST<\/jats:sc>\n            C\n            <jats:sc>UT<\/jats:sc>\n            problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an \u201cintegrality gap instance\u201d for S\n            <jats:sc>PARSEST<\/jats:sc>\n            C\n            <jats:sc>UT<\/jats:sc>\n            . Towards this, we first construct an integrality gap instance for a natural SDP relaxation of U\n            <jats:sc>NIQUE<\/jats:sc>\n            G\n            <jats:sc>AMES<\/jats:sc>\n            . Then we \u201csimulate\u201d the PCP reduction and \u201ctranslate\u201d the integrality gap instance of U\n            <jats:sc>NIQUE<\/jats:sc>\n            G\n            <jats:sc>AMES<\/jats:sc>\n            to an integrality gap instance of S\n            <jats:sc>PARSEST<\/jats:sc>\n            C\n            <jats:sc>UT<\/jats:sc>\n            . This enables us to prove a (log log\n            <jats:italic>n<\/jats:italic>\n            )\n            <jats:sup>1\/6-\u03b4<\/jats:sup>\n            integrality gap for S\n            <jats:sc>PARSEST<\/jats:sc>\n            C\n            <jats:sc>UT<\/jats:sc>\n            , which is known to be equivalent to the metric embedding lower bound.\n          <\/jats:p>","DOI":"10.1145\/2629614","type":"journal-article","created":{"date-parts":[[2015,3,3]],"date-time":"2015-03-03T14:08:19Z","timestamp":1425391699000},"page":"1-39","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":60,"title":["The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into \u2113\n            <sub>1<\/sub>"],"prefix":"10.1145","volume":"62","author":[{"given":"Subhash A.","family":"Khot","sequence":"first","affiliation":[{"name":"Georgia Institute of Technology"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nisheeth K.","family":"Vishnoi","sequence":"additional","affiliation":[{"name":"IBM India Research Lab"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2015,3,2]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1060590.1060675"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2010.59"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.2005.57"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/1374376.1374380"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-007-9007-0"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/1502793.1502794"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539794285983"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-009-9162-6"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2012.83"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539796302531"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02776078"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02785861"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/1361192.1361199"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-006-0210-9"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2010.171.1347"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.5555\/1747597.1748071"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/1132516.1132594"},{"key":"e_1_2_1_18_1","doi-asserted-by":"crossref","volume-title":"Geometry of Cuts and Metrics","author":"Deza M.","DOI":"10.1007\/978-3-642-04295-9"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02589549"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1137\/05064299X"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/129712.129783"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10036"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02614315"},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/227683.227684"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1988.21923"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/2488608.2488610"},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/1597036.1597045"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/509907.510017"},{"key":"e_1_2_1_29_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-005-0745-0"},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2009.v005a004"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2009.37"},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.2005.74"},{"key":"e_1_2_1_33_1","doi-asserted-by":"publisher","DOI":"10.1137\/060660126"},{"key":"e_1_2_1_34_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2006.47"},{"key":"e_1_2_1_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/331524.331526"},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/513400.513441"},{"key":"e_1_2_1_37_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01200757"},{"key":"e_1_2_1_38_1","volume-title":"Lectures on Discrete Geometry. 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