Robust Combinatorial Optimization with Exponential Scenarios

Following the well-studied two-stage optimization framework for stochastic optimization [14, 17], we study approximation algorithms for robust two-stage optimization problems with exponential number of scenarios. Prior to this work, Dhamdhere et al. [7] introduced approximation algorithms for two-stage robust optimization problems with polynomial number of scenarios. To model exponential number of scenarios, we assume the set of possible scenarios is given implicitly, for example by an upper bound on the number of active clients. In two-stage robust optimization, we need to pre-purchase some resources in the first stage before the adversary’s action. In the second stage, after the adversary chooses the clients that need to be covered, we need to complement our solution by purchasing additional resources at an inflated price. The goal is to minimize the cost in the worst-case scenario. We give a general approach for solving such problems using LP rounding. Our approach uncovers an interesting connection between robust optimization and online competitive algorithms. We use this approach, together with known online algorithms, to develop approximation algorithms for several robust covering problems, such as set cover, vertex cover, and edge cover. We also study a simple buy-at-once algorithm that either covers all items in the first round or does nothing in the first round and waits to build the complete solution in the second round. We show that this algorithm gives tight approximation factors for unweighted variants of these covering problems, but performs poorly for general weighted problems. ∗Microsoft Research, Redmond, WA, USA. emails: {urifeige,kamalj,mahdian,mirrokni}@microsoft.com