Results 1  10
of
40
Hedging uncertainty: Approximation algorithms for stochastic optimization problems
 In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization
, 2004
"... We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, ..."
Abstract

Cited by 81 (13 self)
 Add to MetaCart
(Show Context)
We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage 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 primaldual method, and the greedy algorithm, can be carefully adapted to obtain these results. 1
What about Wednesday? approximation algorithms for multistage stochastic optimization
 in APPROX
, 2005
"... Abstract. The field of stochastic optimization studies decision making under uncertainty, when only probabilistic information about the future is available. Finding approximate solutions to wellstudied optimization problems (such as Steiner tree, Vertex Cover, and Facility Location, to name but a f ..."
Abstract

Cited by 34 (9 self)
 Add to MetaCart
(Show Context)
Abstract. The field of stochastic optimization studies decision making under uncertainty, when only probabilistic information about the future is available. Finding approximate solutions to wellstudied optimization problems (such as Steiner tree, Vertex Cover, and Facility Location, to name but a few) presents new challenges when investigated in this framework, which has promoted much research in approximation algorithms. There has been much interest in optimization problems in the setting of twostage stochastic optimization with recourse, which can be paraphrased as follows: On the first day (Monday), we know a probability distribution π from which client demands will be drawn on Tuesday, and are allowed to make preliminary investments (e.g., installing links, opening facilities) towards meeting this future demand. On Tuesday, the actual requirements are revealed (drawn from the same distribution π) and we must purchase enough additional equipment to satisfy these demands; however, these purchases are now made at an inflated cost. In a recent paper [8], we proposed the Boosted Sampling framework which converted an approximation algorithm A for an optimization problem Π into one for the stochastic version of Π (provided A satisfied certain technical conditions). In this paper, we give two generalizations of this Boosted Sampling framework: Firstly, we show that a natural extension of the framework works in a general kstage setting, where information about the future is gradually revealed in several stages and we are allowed to take (increasingly expensive) corrective actions in each stage. We use these to give approximation
How to pay, come what may: Approximation algorithms for demandrobust covering problems
 In FOCS
, 2005
"... Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data or costrobust prob ..."
Abstract

Cited by 31 (9 self)
 Add to MetaCart
(Show Context)
Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data or costrobust problems, we formulate a new “demandrobust” model motivated by recent work on twostage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of firststage solutions for such problems: there exists a firststage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multicut, shortest paths, Steiner trees, vertex cover and uncapacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demandrobust setting.
Sampling bounds for stochastic optimization
 PROC. 9TH RANDOM
, 2005
"... A large class of stochastic optimization problems can be modeled as minimizing an objective function f that depends on a choice of a vector x ∈ X, as well as on a random external parameter ω ∈ Ω given by a probability distribution π. The value of the objective function is a random variable and ofte ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
(Show Context)
A large class of stochastic optimization problems can be modeled as minimizing an objective function f that depends on a choice of a vector x ∈ X, as well as on a random external parameter ω ∈ Ω given by a probability distribution π. The value of the objective function is a random variable and often the goal is to find an x ∈ X to minimize the expected cost Eω[fω(x)]. Each ω is referred to as a scenario. We consider the case when Ω is large or infinite and we are allowed to sample from π in a blackbox fashion. A common method, known as the SAA method (sample average approximation), is to pick sufficiently many independent samples from π and use them to approximate π and correspondingly Eω[fω(x)]. This is one of several scenario reduction methods used in practice. There has been substantial recent interest in twostage stochastic versions of combinatorial optimization problems which can be modeled by the framework described above. In particular, we are interested in the
On twostage stochastic minimum spanning trees
 IN PROC. INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION (IPCO
, 2005
"... We consider the undirected minimum spanning tree problem in a stochastic optimization setting. For the twostage stochastic optimization formulation with finite scenarios, a simple iterative randomized rounding method on a natural LP formulation of the problem yields a nearly bestpossible approxim ..."
Abstract

Cited by 29 (5 self)
 Add to MetaCart
(Show Context)
We consider the undirected minimum spanning tree problem in a stochastic optimization setting. For the twostage stochastic optimization formulation with finite scenarios, a simple iterative randomized rounding method on a natural LP formulation of the problem yields a nearly bestpossible approximation algorithm. We then consider the Stochastic minimum spanning tree problem in a more general blackbox model and show that even under the assumptions of bounded inflation the problem remains log nhard 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 ζ(3)/4 +
Robust combinatorial optimization with exponential scenarios
 In IPCO
, 2007
"... Abstract. Following the wellstudied twostage optimization framework for stochastic optimization [15, 18], we study approximation algorithms for robust twostage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorith ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
(Show Context)
Abstract. Following the wellstudied twostage optimization framework for stochastic optimization [15, 18], we study approximation algorithms for robust twostage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorithms for twostage 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 twostage robust optimization, we need to prepurchase 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 worstcase 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 buyatonce 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. 1
Simple cost sharing schemes for multicommodity rentorbuy and stochastic steiner tree
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... ..."
(Show Context)
Stochastic Steiner Trees without a Root
"... This paper considers the Steiner tree problem in the model of twostage stochastic optimization with recourse. This model, the focus of much recent research [14], tries to capture the fact that many infrastructure planningproblems have to be solved in the presence of uncertainty, and that we have ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
This paper considers the Steiner tree problem in the model of twostage stochastic optimization with recourse. This model, the focus of much recent research [14], tries to capture the fact that many infrastructure planningproblems have to be solved in the presence of uncertainty, and that we have make decisions knowing merely market forecasts (and not the precise set of demands);by the time the actual demands arrive, the costs may be higher due to inflation. In the context of the Stochastic Steiner Tree problem on a graph G = (V, E),the model can be paraphrased thus: on Monday, we are given a probability distribution ss on subsets of vertices, and can build some subset EM of edges. OnTuesday, a set of terminals D materializes (drawn from the same distribution ss).We now have to buy edges ET so that the set EM [ ET forms a Steiner tree on D. The goal is to minimize the expected cost of the solution. We give the first constantfactor approximation algorithm for this problem in thispaper. This is, to the best of our knowledge, the first O(1)approximation forthe stochastic version of a non subadditive problem3 In fact, algorithms for the unrooted stochastic Steiner tree problem we consider in this paper are powerfulenough to solve the Multicommodity RentorBuy problem, themselves a topic of much recent interest [68].
Asking the right questions: Modeldriven optimization using probes
 In Proc. of the 2006 ACM Symp. on Principles of Database Systems
, 2006
"... In several database applications, parameters like selectivities and load are known only with some associated uncertainty, which is specified, or modeled, as a distribution over values. The performance of query optimizers and monitoring schemes can be improved by spending resources like time or bandw ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
(Show Context)
In several database applications, parameters like selectivities and load are known only with some associated uncertainty, which is specified, or modeled, as a distribution over values. The performance of query optimizers and monitoring schemes can be improved by spending resources like time or bandwidth in observing or resolving these parameters, so that better query plans can be generated. In a resourceconstrained situation, deciding which parameters to observe in order to best optimize the expected quality of the plan generated (or in general, optimize the expected value of a certain objective function) itself becomes an interesting optimization problem. We present a framework for studying such problems, and present several scenarios arising in anomaly detection in complex systems, monitoring extreme values in sensor networks, load shedding in data stream systems, and estimating rates in wireless channels and minimum latency routes in networks, which can be modeled in this framework with the appropriate objective functions. Even for several simple objective functions, we show the problems are NpHard. We present greedy algorithms with good performance bounds. The proof of the performance bounds are via novel submodularity arguments.
Pay Today for a Rainy Day: Improved Approximation Algorithms for DemandRobust MinCut and Shortest Path Problems
 STACS
, 2006
"... Abstract. Demandrobust versions of common optimization problems were recently introduced by Dhamdhere et al. [4] motivated by the worstcase considerations of twostage stochastic optimization models. We study the demand robust mincut and shortest path problems, and exploit the nature of the robus ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
(Show Context)
Abstract. Demandrobust versions of common optimization problems were recently introduced by Dhamdhere et al. [4] motivated by the worstcase considerations of twostage stochastic optimization models. We study the demand robust mincut and shortest path problems, and exploit the nature of the robust objective to give improved approximation factors. Specifically, we give a (1 + √ 2) approximation for robust mincut and a 7.1 approximation for robust shortest path. Previously, the best approximation factors were O(log n) for robust mincut and 16 for robust shortest paths, both due to Dhamdhere et al. [4]. Our main technique can be summarized as follows: We investigate each of the second stage scenarios individually, checking if it can be independently serviced in the second stage within an acceptable cost (namely, a guess of the optimal second stage costs). For the costly scenarios that cannot be serviced in this way (“rainy days”), we show that they can be fully taken care of in a nearoptimal first stage solution (i.e., by ”paying today”). We also consider “hittingset ” extensions of the robust mincut and shortest path problems and show that our techniques can be combined with algorithms for Steiner multicut and group Steiner tree problems to give similar approximation guarantees for the hittingset versions of robust mincut and shortest path problems respectively. 1