Chapter

Integer Programming and Combinatorial Optimization

Volume 3509 of the series Lecture Notes in Computer Science pp 321-334

On Two-Stage Stochastic Minimum Spanning Trees

  • Kedar DhamdhereAffiliated withDept. of Computer Science, Carnegie Mellon University
  • , R. RaviAffiliated withTepper School of Business, Carnegie Mellon University
  • , Mohit SinghAffiliated withTepper School of Business, Carnegie Mellon University

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Abstract

We consider the undirected minimum spanning tree problem in a stochastic optimization setting. For the two-stage stochastic optimization formulation with finite scenarios, a simple iterative randomized rounding method on a natural LP formulation of the problem yields a nearly best-possible approximation algorithm.

We then consider the Stochastic minimum spanning tree problem in a more general black-box model and show that even under the assumptions of bounded inflation the problem remains log n-hard to approximate unless P = NP; where n is the size of graph. We also give approximation algorithm matching the lower bound up to a constant factor.

Finally, we consider a slightly different cost model where the second stage costs are independent random variables uniformly distributed between [0,1]. We show that a simple thresholding heuristic has cost bounded by the optimal cost plus \(\frac{\zeta(3)}{4}+o(1)\).