Abstract. Following the well-studied two-stage optimization framework for stochastic optimization [15, 18], 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 buyat-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. 1

This paper studies an extension of the k-median problem where we are given a metric space (V, d) and not just one but m client sets {Si ⊆ V} m i=1, and the goal is to open k facilities F to minimize: maxi∈[m] j∈Si d(j, F) �, i.e., the worst-case cost over all the client sets. This is a “min-max ” or “robust ” version of the k-median problem; however, note that in contrast to previous papers on robust/stochastic problems, we have only one stage of decision-making—where should we place the facilities? We present an O(log n+log m) approximation for robust k-median: The algorithm is combinatorial and very simple, and is based on reweighting/Lagrangeanrelaxation ideas. In fact, we give a general framework for (minimization) facility location problems where there is a bound on the number of open facilities. For robust and stochastic versions of such location problems, we show that if the problem satisfies a certain “projection” property, essentially the same algorithm gives a logarithmic approximation ratio in both versions. We use our framework to give the first approximation algorithms for robust/stochastic versions of k-tree, capacitated k-median, and fault-tolerant k-median. 1

We study two-stage robust variants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously

doi 10.1287/moor.1090.0428

We consider various stochastic models that incorporate the notion of risk-averseness into the standard 2-stage recourse model, and develop novel techniques for solving the algorithmic problems arising in these models. A key notable feature of our work that distinguishes it from work in some other related models, such as the (standard) budget model and the (demand-) robust model, is that we obtain results in the black-box setting, that is, where one is given only sampling access to the underlying distribution. Our first model, which we call the risk-averse budget model, incorporates the notion of risk-averseness via a probabilistic constraint that restricts the probability (according to the underlying distribution) with which the second-stage cost may exceed a given budget B to at most a given input threshold ρ. We also a consider a closely-related model that we call the risk-averse robust model, where we seek to minimize the first-stage cost and the (1 − ρ)-quantile (according to the distribution) of the second-stage cost. We obtain approximation algorithms for a variety of combinatorial optimization problems including the set cover, vertex cover, multicut on trees, min cut, and facility location problems, in the risk-averse budget and robust models with black-box distributions. We first devise a fully polynomial approximation scheme for solving the LP-relaxations of a wide-variety of risk-averse budgeted problems. Complementing

My research in applied algorithms spans the areas of approximation algorithms, online algorithms, data structures, machine learning, and electronic commerce. My recent work has focused on two broad research agendas, namely uniquely represented data structures, and optimization under uncertainty. Uniquely Represented Data Structures. An implementation of an abstract data type (ADT) on a machine model is uniquely represented if it encodes each ADT state with a unique machine state. For my Ph.D. thesis I developed the first efficient constructions of uniquely represented hash tables, linked lists, binary search trees, and many other data structures on the RAM model of computation [3, 4, 7]. These results reversed over thirty years of pessimism in the research literature [1, 5, 13, 18]. These data structures provide the foundation for strongly history independent systems that store exactly the information specified by their designs, such as a filesystem that can delete a file in such a way that there is provably no trace on the system that the file ever existed, at least at the level of the machine model (i.e., bits rather than magnetic moments). They also provide the basis for fast equality testing of complex objects which may prove useful for speeding up software verification; I intend to explore this possibility in future work. Optimization Under Uncertainty. I have developed online algorithms [15, 16, 14] for some general classes of optimization problems, for example monotone submodular function maximization subject to a budget