{"id":324257,"date":"2016-11-18T14:43:05","date_gmt":"2016-11-18T22:43:05","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&p=324257"},"modified":"2018-10-16T20:37:14","modified_gmt":"2018-10-17T03:37:14","slug":"capacitated-network-design-undirected-graphs","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/capacitated-network-design-undirected-graphs\/","title":{"rendered":"Capacitated Network Design on Undirected Graphs"},"content":{"rendered":"

In this paper, we study the approximability of the capacitated network design problem<\/em> (Cap-NDP) on undirected<\/em> graphs: Given G<\/em>\u2009=\u2009(V<\/em>,E<\/em>) with non-negative costs c<\/em> and capacities u<\/em> on its edges, source-sink pairs (s<\/em> i<\/em> <\/sub>, t<\/em> i<\/em> <\/sub>) with demand r<\/em> i<\/em> <\/sub>, the goal is to find the minimum cost subgraph where the minimum (s<\/em> i<\/em> <\/sub>, t<\/em> i<\/em> <\/sub>) cut with u<\/em>-capacities is at least r<\/em> i<\/em> <\/sub>. When u<\/em>\u2009\u2261\u20091, we get the usual SNDP for which Jain gave a 2-approximation algorithm [9]. Prior to our work, the approximability of undirected Cap-NDP was not well understood even in the single source-sink pair case. In this paper, we show that the single-source pair Cap-NDP is label-cover hard in undirected graphs.<\/p>\n

An important special case of single source-sink pair undirected Cap-NDP is the following source location problem<\/em>. Given an undirected graph, a collection of sources S<\/em> and a sink t<\/em>, find the minimum cardinality subset S<\/em>\u2032\u2009\u2286\u2009S<\/em> such that flow(S<\/em>\u2032,t<\/em>), the maximum flow from S<\/em>\u2032 to t<\/em>, equals flow(S<\/em>,t<\/em>). In general, the problem is known to be set-cover hard. We give a O<\/em>(\u03c1<\/em>)-approximation when flow(s<\/em>,t<\/em>)\u2009\u2248\u2009 \u03c1<\/em> <\/sub>flow(s<\/em>\u2032,t<\/em>) for s<\/em>, s<\/em>\u2032\u2009\u2208\u2009S<\/em>, that is, all sources have max-flow values to the sink within a multiplicative \u03c1<\/em> factor of each other.<\/p>\n

The main technical ingredient of our algorithmic result is the following theorem which may have other applications. Given a capacitated, undirected graph G<\/em> with a dedicated sink t<\/em>, call a subset X<\/em>\u2009\u2286\u2009V<\/em> irreducible<\/em> if the maximum flow f<\/em>(X<\/em>) from X<\/em> to t<\/em> is strictly greater than that from any strict subset X<\/em>\u2032\u2009\u2282\u2009X<\/em>, to t<\/em>. We prove that for any irreducible set, X<\/em>, the flow f<\/mi>(<\/mo>X<\/mi>)<\/mo>≥<\/mo>1<\/mn>2<\/mn><\/mfrac>∑<\/mo>i<\/mi>∈<\/mo>X<\/mi><\/mrow><\/munder>f<\/mi>i<\/mi><\/msub><\/math>\">f<\/span>(<\/span>X<\/span>)<\/span>\u2265<\/span>1\/<\/span>2<\/span><\/span>\u2211<\/span>i<\/span>\u2208<\/span>X<\/span><\/span><\/span><\/span>f<\/span>i<\/span><\/span><\/span><\/span><\/span><\/span>, where f<\/em> i<\/em> <\/sub>is the max-flow from i<\/em> to t<\/em>. That is, undirected flows are quasi-additive on irreducible sets.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this paper, we study the approximability of the capacitated network design problem (Cap-NDP) on undirected graphs: Given G\u2009=\u2009(V,E) with non-negative costs c and capacities u on its edges, source-sink pairs (s i , t i ) with demand r i , the goal is to find the minimum cost subgraph where the minimum (s […]<\/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":"Springer Berlin Heidelberg","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, 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