Node-weighted Prize-collecting Survivable Network Design Problems

We consider node-weighted network design problems, in particular the survivable network design problem (SNDP) and its prize-collecting version (PC-SNDP). The input consists of a node-weighted undirected graph G = (V,E) and integral connectivity requirements r(st) for each pair of nodes st. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair st, H contains r(st) disjoint paths between s and t. PC-SNDP is a generalization in which the input also includes a penalty π(st) for each pair, and the goal is to find a subgraph H to minimize the sum of the weight of H and the sum of the penalties for all pairs whose connectivity requirements are not fully satisfied by H . We consider three types of connectivity requirements, edge-connectivity (EC), elementconnectivity (ELC) and vertex-connectivity (VC). Let k = maxst r(st) be the maximum requirement. There has been no non-trivial approximation for node-weighted PC-SNDP for k > 1 even in edge-connectivity setup. We describe multiroute-flow based relaxations for PC-EC-SNDP and PC-ELC-SNDP and obtain approximation algorithms for PC-SNDP and PC-ELC-SNDP through them. The approximation ratios we obtain for PC-EC-SNDP are similar to those that were previously known for EC-SNDP via combinatorial algorithms. Specifically, for PC-EC-SNDP (and PC-ELC-SNDP) we obtain an O(k log n)-approximation in general graphs and an O(k)-approximation in graphs that exclude a fixed minor. Moreover, based on the approximation algorithm of ELC-SNDP and the reduction method of Chuzhoy and Khanna [6] we obtain O(k log n)-approximation for PC-VC-SNDP which improves to O(k log n) on instances from a minor-closed families of graphs.

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