Thresholded Covering Algorithms for Robust and Max-min Optimization

In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the k-robust model where the possible scenarios tomorrow are given by all demand-subsets of size k. In this paper, we give the following simple and intuitive template for k-robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for k-robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our k-robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, An extended abstract containing the results of this paper and of [27] appeared jointly in Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), 2010. A. Gupta: Supported in part by NSF awards CCF-0448095 and CCF-0729022, and an Alfred P. Sloan Fellowship. R. Ravi: Supported in part by NSF Grants CCF-0728841 and CCF-1218382. A. Gupta Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA e-mail: [email protected] V. Nagarajan (B) IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA e-mail: [email protected] R. Ravi Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA e-mail: [email protected]