{"id":185836,"date":"2010-05-12T00:00:00","date_gmt":"2011-01-13T12:37:42","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/finding-needles-in-exponential-haystacks\/"},"modified":"2016-08-22T11:28:21","modified_gmt":"2016-08-22T18:28:21","slug":"finding-needles-in-exponential-haystacks","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/finding-needles-in-exponential-haystacks\/","title":{"rendered":"Finding Needles in Exponential Haystacks"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Let S<sub>1<\/sub>,&#8230;,S<sub>n<\/sub> be subsets of 1,&#8230;n.  A quarter century ago I showed that there is a two-coloring of the vertices so that all sets have discrepancy at most K&radic;n, K an absolute constant.  The colors  &chi;(j) are +1,-1 and discrepancy is the absolute value of the sum of the colors. In matrix terms, given an n by n matrix A with all coefficients in [-1,+1] there is a vector x with values -1,+1 so that Ax has L-infinity norm at most K&radic;n].) I had long conjectured that there would not be an algorithm to find such a coloring.  Very recently Nikhil Bansal (IBM) has found an algorithm, which I shall describe.  At its heart it uses Semidefinite Programming.  The  colors &chi;(j) &#8220;float&#8221; in [-1,+1], each performing a discretized Brownian motion until being caught by the boundary. Unlike the recent Moser-Tardos work on the Lovasz Local Lemma, this does not provide a new proof \u2013 the known arguments are used to prove feasibility of certain Semidefinite Programs.  Along the way, this provides a fresh view of the original argument via a &#8220;cost equation.&#8221;<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let S1,&#8230;,Sn be subsets of 1,&#8230;n. A quarter century ago I showed that there is a two-coloring of the vertices so that all sets have discrepancy at most K&radic;n, K an absolute constant. The colors &chi;(j) are +1,-1 and discrepancy is the absolute value of the sum of the colors. In matrix terms, given an [&hellip;]<\/p>\n","protected":false},"featured_media":280919,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-185836","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/onhGG2d48I0","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185836"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185836\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/280919"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185836"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185836"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185836"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185836"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185836"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185836"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}