Theory Comput Syst DOI 10.1007/s00224-014-9548-6 Approximation Algorithms for Fragmenting a Graph Against a Stochastically-Located Threat David B. Shmoys · Gwen Spencer © Springer Science+Business Media New York 2014 Abstract Motivated by issues in allocating limited preventative resources to pro- tect a landscape against the spread of a wildfire from a stochastic ignition point, we give approximation algorithms for a new family of stochastic optimization problems. We study several models in which we are given a graph with edge costs and node values, a budget, and a probabilistic distribution over ignition nodes: the goal is to find a budget-limited set of edges whose removal protects the largest expected value from being reachable from a stochastic ignition node. In particular, 2-stage stochastic models capture the tradeoffs between preventative treatment and real-time response. The resulting stochastic cut problems are interesting in their own right, and cap- ture a number of related interdiction problems, both in the domain of computational sustainability, and beyond. In trees, even the deterministic problem is (weakly) NP hard: we give a Polynomial-time approximation scheme for the single-stage stochas- tic case in trees when the number of scenarios is constant. For the 2-stage stochastic model in trees we give a (1 − 1 e )<(1 − (1 − 1/2δ) 2δ )-approximation in trees which violates the budget by a factor of at most 2 (δ is the tree diameter), and a 0.387- approximation that is budget-balanced. For the single-stage stochastic case in trees we can save (1 − (1 − 1/δ) δ )OPT without violating the budget. Single-stage results extend to general graphs with an additional O(log n) loss in budget-balancedness. Multistage results have a similar extension when the number of scenarios is con- stant. In an extension of the single-stage model where both ignition and spread are A preliminary version of this work appeared in the Proceedings of the Workshop on Approximation and Online Algorithms, 2011. D. B. Shmoys School of ORIE and Department of Computer Science, Cornell University, Ithaca, NY, 14853, USA e-mail: shmoys@cs.cornell.edu G. Spencer () Neukom Institute, Dartmouth College, Hanover, NH, 03755, USA e-mail: gwenspencer@gmail.com