{"id":184013,"date":"2005-08-18T00:00:00","date_gmt":"2009-10-31T13:17:53","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/approximability-of-the-unique-coverage-problem\/"},"modified":"2016-09-09T09:48:54","modified_gmt":"2016-09-09T16:48:54","slug":"approximability-of-the-unique-coverage-problem","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/approximability-of-the-unique-coverage-problem\/","title":{"rendered":"Approximability of the Unique Coverage Problem"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In this talk, we consider a natural maximization version of the well-known set cover problem, called unique coverage: given a collection of sets, find a sub-collection that maximizes the number of elements covered exactly once. This problem, which is motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, and radio broadcasting.  We prove sub-logarithmic hardness results for unique coverage.  Specifically, we prove <i>&Omega;(log<sup>&sigma;(&epsilon;)<\/sup> n)<\/i> inapproximability assuming that <i>NP  not &sube; BPTIME (2<sup>n^&epsilon;<\/sup>)<\/i> for some <i>&epsilon; > 0<\/i>.  We also prove <i>&Omega;(log<sup>1\/3-&epsilon;<\/sup> n)<\/i> inapproximability, for any <i>&epsilon; > 0<\/i>, assuming that refuting random instances of 3SAT is hard on average; and prove <i>&Omega;(log  n)<\/i> inapproximability under a plausible hypothesis. We establish easy logarithmic upper bounds even for a more general (budgeted) setting, giving an <i>O (log  B)<\/i> approximation algorithm when every set has at most <i>B<\/i> elements.  Our hardness results have other implications regarding the hardness of some well-studied problems in computational economics.  In particular, for the problem of unlimited-supply single-minded (envy-free) pricing, a recent result by Guruswami et al. [SODA&#8217;05] proves an <i>O (log  n)<\/i> approximation, but no inapproximability result better than APX-hardness is known.  Our hardness results for the unique coverage problem imply the same hardness of approximation bounds for this version of envy-free pricing.<\/p>\n<p>Joint work with: E. Demaine, U. Feige, and M. Hajiaghayi.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this talk, we consider a natural maximization version of the well-known set cover problem, called unique coverage: given a collection of sets, find a sub-collection that maximizes the number of elements covered exactly once. This problem, which is motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, [&hellip;]<\/p>\n","protected":false},"featured_media":195279,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-184013","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/gFFBwsRBDeQ","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184013","targetHints":{"allow":["GET"]}}],"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\/184013\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195279"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184013"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184013"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184013"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184013"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184013"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184013"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184013"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184013"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184013"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184013"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}