For Peer Review Two - stage Robust Network Design with Exponential Scenarios

We study two-stage robust variants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by Dhamdhere et al. [9], Golovin et al. [15], and Feige et al. [13], are two-stage planning problems in which the requirements are revealed after some decisions are taken in stage one. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust Steiner tree problem, for example, one buys some edges in stage one after which some terminals are revealed. In the second stage, one has to buy more edges, at a higher cost, to complete the stage one solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially many scenarios given implicitly. A scenario consists of any subset of k terminals (for Steiner tree), or any subset of k terminal-pairs (for Steiner forest), or any subset of k clients (for facility location). Feige et al. [13] give an LP-based framework for approximation algorithms for for a class of covering problems two stage robust problems, but their framework does not work for network design problems like Steiner tree, and gives only a logarithmic approximation for the robust facility location problem. We present the first constant-factor approximation algorithms for the robust Steiner tree (with exponential number of scenarios) and robust uncapacitated facility location problems. Our algorithms are combinatorial and are based on guessing the optimum cost and clustering to aggregate near-by vertices. For the robust Steiner forest problem on trees and with uniform inflation, we present a constant approximation and show that the problem on general graphs and with two inflation factors is hard to approximate within O(log n) factor, for any constant ǫ > 0, unless NP has randomized quasi-polynomial time algorithms. Finally, we showAPX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question by [9] and [15].