{"id":166202,"date":"2014-01-01T00:00:00","date_gmt":"2014-01-01T00:00:00","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/msr-research-item\/cops-robbers-and-threatening-skeletons-padded-decomposition-for-minor-free-graphs\/"},"modified":"2018-10-16T20:11:38","modified_gmt":"2018-10-17T03:11:38","slug":"cops-robbers-and-threatening-skeletons-padded-decomposition-for-minor-free-graphs","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/cops-robbers-and-threatening-skeletons-padded-decomposition-for-minor-free-graphs\/","title":{"rendered":"Cops, Robbers, and Threatening Skeletons: Padded Decomposition for Minor-Free Graphs"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We prove that any graph excluding <i>K<sub>r<\/sub><\/i> as a minor has can be partitioned into clusters of diameter at most <i>\u0394<\/i> while removing at most <i>O(r\/\u0394)<\/i> fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove <i>O(r<sup>2<\/sup>\/\u0394)<\/i> fraction of the edges.<\/p>\n<p>Our result is obtained by a new approach to relate the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops-and-robbers game on excluded-minor graphs can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter <i>O(r)<\/i> and strong-diameter partitions with padding parameter <i>O(r<sup>2<\/sup>)<\/i> for <i>K<sub>r<\/sub><\/i>-free graphs, padding <i>O(k)<\/i> for graphs with treewidth <i>k<\/i>, and padding <i>O(log g)<\/i> for graphs with genus <i>g<\/i>.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most \u0394 while removing at most O(r\/\u0394) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to [&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":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"STOC '14 Proceedings of the 46th symposium on Theory of Computing","msr_chapter":"","msr_edition":"STOC, STOC '14 Proceedings of the 46th symposium on Theory of Computing","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"STOC '14 Proceedings of the 46th symposium on Theory of 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