{"id":356267,"date":"2017-01-20T13:15:42","date_gmt":"2017-01-20T21:15:42","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=356267"},"modified":"2018-10-16T19:58:25","modified_gmt":"2018-10-17T02:58:25","slug":"finding-hidden-cliques-linear-time-high-probability","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/finding-hidden-cliques-linear-time-high-probability\/","title":{"rendered":"Finding Hidden Cliques in Linear Time with High Probability"},"content":{"rendered":"<p>We are given a graph <em>G<\/em> with <em>n<\/em> vertices, where a random subset of <em>k<\/em> vertices has been made into a clique, and the remaining edges are chosen independently with probability \u00bd . This random graph model is denoted <em>G<\/em>(<em>n,<\/em> \u00bd , <em>k<\/em>). The hidden clique problem is to design an algorithm that finds the <em>k<\/em>-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov uses spectral techniques to find the hidden clique with high probability when <em>k<\/em> = <em>c\u221an<\/em> for a sufficiently large constant <em>c<\/em> > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron. It has the advantages of being simpler and more intuitive, and of an improved running time of<em> O<\/em>(<em>n<\/em><sup>2<\/sup>). However, the analysis in the paper gives success probability of only 2\/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time <em>O<\/em>(<em>n<\/em><sup>2<\/sup>), and has a failure probability that is less than polynomially small.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability \u00bd . This random graph model is denoted G(n, \u00bd , k). The hidden clique problem is to design an algorithm that finds the [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-356267","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cornell University Library","msr_edition":"","msr_affiliation":"","msr_published_date":"2010-10-14","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":"356273","msr_publicationurl":"http:\/\/arxiv.org\/abs\/arXiv:1010.2997","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"1010.2997","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2017\/01\/1010.2997.pdf","id":356273,"label_id":0},{"type":"url","title":"http:\/\/arxiv.org\/abs\/arXiv:1010.2997","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"http:\/\/arxiv.org\/abs\/arXiv:1010.2997"}],"msr-author-ordering":[{"type":"text","value":"Yael Dekel","user_id":0,"rest_url":false},{"type":"text","value":"Ori Gurel-Gurevich","user_id":0,"rest_url":false},{"type":"user_nicename","value":"peres","user_id":33234,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=peres"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[392777],"publication":[],"video":[],"download":[],"msr_publication_type":"article","related_content":{"projects":[{"ID":392777,"post_title":"Foundations of Optimization","post_name":"foundations-of-optimization","post_type":"msr-project","post_date":"2017-07-06 09:30:53","post_modified":"2018-12-04 14:12:39","post_status":"publish","permalink":"https:\/\/www.microsoft.com\/en-us\/research\/project\/foundations-of-optimization\/","post_excerpt":"Optimization methods are the engine of machine learning algorithms. Examples abound, such as training neural networks with stochastic gradient descent, segmenting images with submodular optimization, or efficiently searching a game tree with bandit algorithms. We aim to advance the mathematical foundations of both discrete and continuous optimization and to leverage these advances to develop new algorithms with a broad set of AI applications. 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