{"id":185065,"date":"2010-08-17T00:00:00","date_gmt":"2010-08-19T13:45:58","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/finding-all-min-st-cuts-in-planar-graphs\/"},"modified":"2016-08-22T11:28:15","modified_gmt":"2016-08-22T18:28:15","slug":"finding-all-min-st-cuts-in-planar-graphs","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/finding-all-min-st-cuts-in-planar-graphs\/","title":{"rendered":"Finding all min st-cuts in planar graphs"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In 1961, Gomory and Hu observed that the minimum st-cuts in an edge-weighted, undirected graph, for all pairs of vertices, can be represented compactly by a tree: the GH-tree.  Not only that, they showed that one can find this tree with only n-1 minimum cut computations.  Besides improvements to max flow\/min cut algorithms, not much progress has been made on the all-pairs min-cut problem:  Gusfield (&#8217;90) simplified Gomory and Hu&#8217;s algorithm (without improvements the asymptotic running time) and Balghat et. al. (&#8217;07) showed how to find a GH-tree for unweighted graphs in  time faster than required for n-1 min cut computations.<\/p>\n<p>Here we study the problem in planar graphs.  Using the duality of cuts and cycles and small, balanced separators, we show how to find a GH-tree for weighted, undirected planar graphs in O(n polylog n) time.<\/p>\n<p>To appear in FOCS&#8217;10. Joint work with Piotr Sankowski and Christian Wulff-Nilsen<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 1961, Gomory and Hu observed that the minimum st-cuts in an edge-weighted, undirected graph, for all pairs of vertices, can be represented compactly by a tree: the GH-tree. Not only that, they showed that one can find this tree with only n-1 minimum cut computations. Besides improvements to max flow\/min cut algorithms, not much [&hellip;]<\/p>\n","protected":false},"featured_media":195686,"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-185065","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/MneGZIXwnS4","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185065","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\/185065\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195686"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185065"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185065"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185065"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185065"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=185065"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=185065"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185065"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185065"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=185065"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=185065"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}