{"id":393056,"date":"2017-04-10T00:00:36","date_gmt":"2017-04-10T07:00:36","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=393056"},"modified":"2018-10-16T20:12:54","modified_gmt":"2018-10-17T03:12:54","slug":"local-max-cut-smoothed-polynomial-time","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/local-max-cut-smoothed-polynomial-time\/","title":{"rendered":"Local Max-Cut In Smoothed Polynomial Time"},"content":{"rendered":"<p>In 1988, Johnson, Papadimitriou and Yannakakis wrote that &#8220;Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems&#8221;. Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and indeed complete) example is the local max-cut problem, for which no polynomial time method is known. In a breakthrough paper, Etscheid and R\\&#8221;oglin proved that the smoothed complexity of local max-cut is quasi-polynomial, i.e., if arbitrary bounded weights are randomly perturbed, a local maximum can be found in <em>n<sup>O(<\/sup><\/em><sup>logn<\/sup><em><sup>)<\/sup><\/em> steps. In this paper we prove smoothed polynomial complexity for local max-cut, thus confirming that finding local optima for max-cut is much easier than solving it.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 1988, Johnson, Papadimitriou and Yannakakis wrote that &#8220;Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems&#8221;. Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and indeed complete) example is [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13546],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-393056","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2017-04-10","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"393059","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1610.04807","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"1610.04807","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2017\/06\/1610.04807.pdf","id":393059,"label_id":0},{"type":"url","title":"https:\/\/arxiv.org\/abs\/1610.04807","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1610.04807"}],"msr-author-ordering":[{"type":"text","value":"Omer 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