Robust Combinatorial Optimization with Exponential Scenarios

  • Uriel Feige
  • Kamal Jain
  • Mohammad Mahdian
  • Vahab Mirrokni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4513)


Following the well-studied two-stage optimization framework for stochastic optimization  [15,8], we study approximation algorithms for robust two-stage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorithms for two-stage robust optimization problems with explicitly given scenarios. In this paper, 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 stage or does nothing in the first stage and waits to build the complete solution in the second stage. We show that this algorithm gives tight approximation factors for unweighted variants of these covering problems, but performs poorly for general weighted problems.


Approximation Algorithm Robust Optimization Online Algorithm Vertex Cover Vertex Cover Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Uriel Feige
    • 1
  • Kamal Jain
    • 1
  • Mohammad Mahdian
    • 2
  • Vahab Mirrokni
    • 1
  1. 1.Microsoft Research 
  2. 2.Yahoo! Research 

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