Improved approximations for two-stage min-cut and shortest path problems under uncertainty

In this paper, we study the robust and stochastic versions of the two-stage mincut and shortest path problems introduced in Dhamdhere et al. [6], and give approximation algorithms with improved approximation factors. Specifically, we give a 2-approximation for the robust min-cut problem and a 4-approximation for the stochastic version. For the two-stage shortest path problem, we give a 3.39-approximation for the robust version and 6.78-approximation for the stochastic version. Our results significantly improve the previous best approximation factors for the problems. In particular, we provide the first constantfactor approximation for the stochastic min-cut problem. Our algorithms are based on guess and prune strategy that crucially exploits the nature of the robust and stochastic objective. In particular, we guess the worst-case second stage cost and based on the guess, select a subset of costly scenarios for the first-stage solution to address. The second-stage solution for any scenario is simply the min-cut (or shortest Part of this work was done while D. Golovin was affiliated with Carnegie Mellon University and was supported by NSF grants CCR-0122581 and IIS-0121678. V. Goyal is supported by NSF grant CMMI-120116. R. Ravi is supported by NSF grants CCF-1218382 and CCF-1143998. V. Polishchuk is supported by the Academy of Finland grant 138520. M. Sysikaski is supported by research funds of University of Helsinki. D. Golovin Google, Inc. E-mail: [email protected] V. Goyal Department of Industrial Engineering and Operations Research Columbia University E-mail: [email protected] V. Polishchuk University of Helsinki E-mail: [email protected] R. Ravi Tepper School of Business Carnegie Mellon University E-mail: [email protected] M. Sysikaski University of Helsinki E-mail: [email protected]