{"id":164485,"date":"2013-06-01T00:00:00","date_gmt":"2013-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/approaching-the-chasm-at-depth-four\/"},"modified":"2018-10-16T20:19:28","modified_gmt":"2018-10-17T03:19:28","slug":"approaching-the-chasm-at-depth-four","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/approaching-the-chasm-at-depth-four\/","title":{"rendered":"Approaching the chasm at depth four"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Agrawal-Vinay (FOCS 2008) and Koiran (TCS 2012) have recently shown that an exp(\u221a(n) * log<sup>2<\/sup> n))  lower bound for depth four homogeneous circuits computing the permanent with bottom layer of gates having fanin bounded by \u221a(n)  translates to super-polynomial lower bound for a general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin.<\/p>\n<p>We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by \u221a(n)  computing the permanent (or the determinant) must be of size exp(\u221a(n)) .<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Agrawal-Vinay (FOCS 2008) and Koiran (TCS 2012) have recently shown that an exp(\u221a(n) * log2 n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of gates having fanin bounded by \u221a(n) translates to super-polynomial lower bound for a general arithmetic circuits computing the permanent. Motivated by this, we examine 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":[13561],"msr-publication-type":[193716],"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-164485","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"IEEE","msr_edition":"Conference on Computational 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Award","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":"205422","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"lowerbound.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/lowerbound-1.pdf","id":205422,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":205422,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/lowerbound-1.pdf"}],"msr-author-ordering":[{"type":"user_nicename","value":"ankgu","user_id":31033,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=ankgu"},{"type":"text","value":"Pritish Kamath","user_id":0,"rest_url":false},{"type":"user_nicename","value":"neeraka","user_id":33076,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=neeraka"},{"type":"text","value":"Ramprasad 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