Node-Weighted Network Design in Planar and Minor-Closed Families of Graphs
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We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph G=(V,E) and integer connectivity requirements r(uv) for each unordered pair of nodes uv. The goal is to find a minimum weighted subgraph H of G such that H contains r(uv) disjoint paths between u and v for each node pair uv. Three versions of the problem are edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP) and vertex-connectivity SNDP (VC-SNDP) depending on whether the paths are required to be edge, element or vertex disjoint respectively. Our main result is an O(k)-approximation algorithm for EC-SNDP and Elem-SNDP when the input graph is planar or more generally if it belongs to a proper minor-closed family of graphs; here k=max_uv r(uv) is the maximum connectivity requirement. This improves upon the O(k log n)-approximation known for node-weighted EC-SNDP and Elem-SNDP in general graphs [Nutov, TALG'12]. We also obtain an O(1) approximation for node-weighted VC-SNDP when the connectivity requirements are in {0,1,2}; for higher connectivity our result for Elem-SNDP can be used in a black-box fashion to obtain a logarithmic factor improvement over currently known general graph results. Our results are inspired by, and generalize, the work of [Demaine, Hajiaghayi and Klein, TALG'14] who obtained constant factor approximations for node-weighted Steiner tree and Steiner forest problems in planar graphs and proper minor-closed families of graphs via a primal-dual algorithm.
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