Online and stochastic survivable network design

Consider the edge-connectivity survivable network design problem: given a graph G = (V,E) with edge-costs, and edge-connectivity requirements r _ij ε ℤ_≥0 for every pair of vertices i, j ε V , find an (approximately) minimum-cost net- work that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(r_max log ³ n) competitive online algorithm for this edge-connectivity network design problem, where r_max = max_ij r_ij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected sub- graphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approxima- tion algorithms that admit strict cost-shares with the same strictness value. This, in turn, implies approximation al-gorithms for (a) the rent-or-buy version and (b) the (two- stage) stochastic version of the edge-connected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edge-costs are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)-strict cost shares, which gives constant-factor rent-or-buy and stochastic algorithms for these instances.