## Hedging uncertainty: Approximation algorithms for stochastic optimization problems (2004)

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Venue: | In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization |

Citations: | 67 - 10 self |

### BibTeX

@INPROCEEDINGS{Ravi04hedginguncertainty:,

author = {R. Ravi and Amitabh Sinha},

title = {Hedging uncertainty: Approximation algorithms for stochastic optimization problems},

booktitle = {In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization},

year = {2004},

pages = {101--115}

}

### Years of Citing Articles

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### Abstract

We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of two-stage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be carefully adapted to obtain these results. 1

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Citation Context ...mber of combinatorial elements (the number of vertices in the shortest paths problems and the number of elements 3sProblem Det. Stochastic Stochastic Hardness approx. elements approximation Shortest 1=-=[9]-=- Sink O(1) MAX-SNP paths Sink and metric O(log 2 n log m) Ω(log 2 n) Bin packing APTAS[7] Object sizes APTAS NP-complete[7] Facility 1.52[26] Client demands, 8 1.46[12] location facility costs Vertex ... |

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Citation Context ...ochastic optimization problems with a restriction on the cost function: in the second stage, all costs go up uniformly by a factor of λ. A generalization of their model was considered by Gupta et al. =-=[12]-=-, who provided approximation algorithms with only sampling access to the second-stage realization process, thereby obtaining a framework which can handle an arbitrary number of scenariosaslongastherei... |

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Citation Context ..., most approximation algorithms assume complete knowledge of the input at the outset, barring a few exceptions such as scheduling problems [26,32]. Recently, and independently of us, Immorlica et al. =-=[15]-=- considered approximation algorithms for stochastic optimization problems with a restriction on the cost function: in the second stage, all costs go up uniformly by a factor of λ. A generalization of ... |

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Citation Context ...l optimization problems. Gupta, Ravi and Sinha [16] considered the two-stage finite-scenario version of the rooted Steiner tree problem, and provided a constant factor approximation. Shmoys and Swamy =-=[39, 40]-=- provide a sampling-based approach which uses interior point linear programming algorithms to provide approximation algorithms for two-stage as well as multi-stage stochastic versions of a certain cla... |

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Citation Context ...times of future jobs; some approximation algorithms which account for such uncertainty in various models of scheduling include the works of Möhring, Schulz and Uetz [33], Kleinberg, Tardos and Rabani =-=[27]-=-, and Skutella and Uetz [43]. Karger and Minkoff [26] considered a Steiner tree problem with uncertainty in the terminal set, which falls under a slightly different model of stochastic optimization. K... |

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Citation Context ...int linear programming algorithms to provide approximation algorithms for two-stage as well as multi-stage stochastic versions of a certain class of combinatorial optimization problems. Gupta, et al. =-=[14, 15]-=- also provide sampling-based approximation algorithms for two-stage and multi-stage stochastic problems, using cost-sharing functions to bound the cost of the solution. Both of these streams of work a... |

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Citation Context ...l optimization problems. Gupta, Ravi and Sinha [16] considered the two-stage finite-scenario version of the rooted Steiner tree problem, and provided a constant factor approximation. Shmoys and Swamy =-=[39, 40]-=- provide a sampling-based approach which uses interior point linear programming algorithms to provide approximation algorithms for two-stage as well as multi-stage stochastic versions of a certain cla... |

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Citation Context ...urs. Subsequent to the first appearance of this work, other papers have provided further approximation algorithms for stochastic versions of combinatorial optimization problems. Gupta, Ravi and Sinha =-=[16]-=- considered the two-stage finite-scenario version of the rooted Steiner tree problem, and provided a constant factor approximation. Shmoys and Swamy [39, 40] provide a sampling-based approach which us... |

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Citation Context ...stochastic versions of combinatorial optimization problems include that of Hayrapetyan, Swamy and Tardos [20] (information networks), Gupta and Pál [13] (Steiner trees), and Dhamdhere, Ravi and Singh =-=[7]-=- (minimum spanning trees). 2.6. Our results In this paper, we provide polynomial-time approximation algorithms for several classical combinatorial optimization problems, in a two-stage stochastic opti... |

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Citation Context ...tochastic Hardness approx. elements approximation Shortest 1[9] Sink O(1) MAX-SNP paths Sink and metric O(log 2 n log m) Ω(log 2 n) Bin packing APTAS[7] Object sizes APTAS NP-complete[7] Facility 1.52=-=[26]-=- Client demands, 8 1.46[12] location facility costs Vertex cover 2[28] Vertex weights, 2 1.16[15] Incidence Set cover O(log n)[17] Set weights, O(log nm) Ω(log n)[1], Set inclusions Ω(log m) Figure 1:... |

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Citation Context ...of the best approximation algorithm for the bin-packing problem. Any locally optimal algorithm (first-fit, for example) achieves ρ = 2. An asymptotic PTAS was given by Fernandez de la Vega and Lueker =-=[8]-=-, which uses at most (1+2ɛ)OPT +1 bins. The following theorem shows how to extend any bin-packing algorithm to handle stochastic bin-packing. Theorem 7. Order the scenarios so that we have � i s1 � i ... |

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Citation Context ...titutes an improvement over our work. Other recent work providing approximation algorithms for stochastic versions of combinatorial optimization problems include that of Hayrapetyan, Swamy and Tardos =-=[20]-=- (information networks), Gupta and Pál [13] (Steiner trees), and Dhamdhere, Ravi and Singh [7] (minimum spanning trees). 2.6. Our results In this paper, we provide polynomial-time approximation algori... |

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Citation Context ... recent work providing approximation algorithms for stochastic versions of combinatorial optimization problems include that of Hayrapetyan, Swamy and Tardos [20] (information networks), Gupta and Pál =-=[13]-=- (Steiner trees), and Dhamdhere, Ravi and Singh [7] (minimum spanning trees). 2.6. Our results In this paper, we provide polynomial-time approximation algorithms for several classical combinatorial op... |

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Citation Context ...om varying (and typically increased) second-stage facility costs under different scenarios. In the other direction, SFL can be viewed as a special case of the multicommodity facility location problem =-=[37, 41]-=-, where we treat each scenario as a distinct commodity and the cost of a facility depends on the commodities it serves. However, the best-known approximation ratio for such a version of the multicommo... |

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Citation Context ...pproximation algorithms which account for such uncertainty in various models of scheduling include the works of Möhring, Schulz and Uetz [33], Kleinberg, Tardos and Rabani [27], and Skutella and Uetz =-=[43]-=-. Karger and Minkoff [26] considered a Steiner tree problem with uncertainty in the terminal set, which falls under a slightly different model of stochastic optimization. Kong and Schaefer [28] provid... |

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Citation Context ...om varying (and typically increased) second-stage facility costs under different scenarios. In the other direction, SFL can be viewed as a special case of the multicommodity facility location problem =-=[37, 41]-=-, where we treat each scenario as a distinct commodity and the cost of a facility depends on the commodities it serves. However, the best-known approximation ratio for such a version of the multicommo... |