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In this paper, we study the robust and stochastic versions of the two-stage min-cut and shortest path problems introduced in Dhamdhere et al. (in How to pay, come what may: approximation algorithms for demand-robust covering problems. In: FOCS, pp 367–378,  2005 ), and give approximation algorithms with improved approximation factors. Specifically, we give a 2-approximation for the robust min-cut problem and a 4-approximation for the stochastic version. For the two-stage shortest path problem, we give a 3.39 -approximation for the robust version and 6.78 -approximation for the stochastic version. Our results significantly improve the previous best approximation factors for the problems. In particular, we provide the first constant-factor approximation for the stochastic min-cut problem. Our algorithms are based on a guess and prune strategy that crucially exploits the nature of the robust and stochastic objective. In particular, we guess the worst-case second stage cost and based on the guess, select a subset of costly scenarios for the first-stage solution to address. The second-stage solution for any scenario is simply the min-cut (or shortest path) problem in the residual graph. The key contribution is to show that there is a near-optimal first-stage solution that completely satisfies the subset of costly scenarios that are selected by our procedure. While the guess and prune strategy is not directly applicable for the stochastic versions, we show that using a novel LP formulation, we can adapt a guess and prune algorithm for the stochastic versions. Our algorithms based on the guess and prune strategy provide insights about the applicability of this approach for more general robust and stochastic versions of combinatorial problems.

In this paper a class of combinatorial optimization problems is discussed. It is assumed that a feasible solution can be constructed in two stages. In the first stage the objective function costs are known while in the second stage they are uncertain and belong to an interval uncertainty set. In order to choose a solution, the minmax regret criterion is used. Some general properties of the problem are established and results for two particular problems, namely the shortest path and the selection problem, are shown.

In this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second-stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network optimization and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problems, are also provided.

Given a polytope X , a monotone concave univariate function g , and two vectors c and d , we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c ’ x + g ( d ’ x ). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives.

We commence an algorithmic study of Bulk-Robustness , a new model of robustness in combinatorial optimization. Unlike most existing models, Bulk-Robust combinatorial optimization features a highly nonuniform failure model. Instead of an interdiction budget, Bulk-Robust counterparts provide an explicit list of interdiction sets, comprising the admissible set of scenarios, thus allowing to model correlations between failures of different components in the system, interdiction sets of variable cardinality and more. The resulting model is suitable for capturing failures of complex structures in the system. We provide complexity results and approximation algorithms for Bulk-Robust counterparts of the Minimum Matroid Basis problems and the Shortest Path problem. Our results rely on various techniques, and outline the rich and heterogeneous combinatorial structure of Bulk-Robust optimization.

This paper addresses the cooperative task assignment problem for heterogeneous unmanned aerial vehicles with time windows considering uncertain fuel consumption. In the scenario where probabilistic fuel consumption exists and its distribution needs to be estimated from historical data samples, we first formulate the problem as a chance-constrained combinatorial optimization problem and utilize the sample average approximation method to solve it. Further, to address the issue of ambiguous distribution, we introduce distributionally robust chance constraints, which consider a set of probability distributions that are contained within a 1-Wasserstein ball centered around the empirical distribution of field data. We approximate the distributionally robust chance-constrained cooperative task assignment problem by applying a CVaR-based tractable approximation such that the problem can be transformed into a deterministic mixed-integer linear programming problem, which can be efficiently solved by state-of-the-art optimization solvers. Finally, we conduct a series of numerical experiments, which not only verify the computational efficiency of the distributionally robust chance-constrainted models but also reduce the degree of constraint violation in out-of-sample tests compared with a sample average approximation method.

We study the robust assignment problem where the goal is to assign items of various types to containers without exceeding container capacity. We seek an assignment that uses the fewest number of containers and is robust, that is, if any item of type ti becomes corrupt causing the containers with type ti to become unstable, every other item type tj≠ti is still assigned to a stable container. We begin by presenting an optimal polynomial-time algorithm that finds a robust assignment using the minimum number of containers for the case when the containers have infinite capacity. Then we consider the case where all containers have some fixed capacity and give an optimal polynomial-time algorithm for the special case where each type of item has the same size. When the sizes of the item types are nonuniform, we provide a polynomial-time 2-approximation for the problem. We also prove that the approximation ratio of our algorithm is no lower than 1.813. We conclude with an experimental evaluation of our algorithm.

Given a dataset an outlier can be defined as an observation that does not follow the statistical properties of the majority of the data. Computation of the location estimate is of fundamental importance in data analysis, and it is well known in statistics that classical methods, such as taking the sample average, can be greatly affected by the presence of outliers in the data. Using the median instead of the mean can partially resolve this issue but not completely. For the univariate case, a robust version of the median is the Least Trimmed Absolute Deviation (LTAD) robust estimator introduced in Tableman (Stat Probab Lett 19(5):387–398, 1994), which has desirable asymptotic properties such as robustness, consistently, high breakdown and normality. There are different generalizations of the LTAD for multivariate data, depending on the choice of norm. Chatzinakos et al. (J Comb Optim, 2015) we present such a generalization using the Euclidean norm and propose a solution technique for the resulting combinatorial optimization problem, based on a necessary condition, that results in a highly convergent local search algorithm. In this subsequent work, we use the L1 norm to generalize the LTAD to higher dimensions, and show that the resulting mixed integer programming problem has an integral relaxation, after applying an appropriate data transformation. Moreover, we utilize the structure of the problem to show that the resulting LP’s can be solved efficiently using a subgradient optimization approach. The robust statistical properties of the proposed estimator are verified by extensive computational results.

We establish strong NP-hardness and in-approximability of the so-called representatives selection problem, a tool selection problem in the area of robust optimization. Our results answer a recent question of Dolgui and Kovalev (4OR Q J Oper Res 10:181–192, 2012 ).

In this paper we study the weighted tardiness job-shop scheduling problem, taking into consideration the presence of random shop disturbances. A basic thesis of the paper is that global scheduling performance is determined primarily by a subset of the scheduling decisions to be made. By making these decisions in an a priori static fashion, which maintains a global perspective, overall performance efficiency can be achieved. Further, by allowing the remaining decisions to be made dynamically, flexibility can be retained in the schedule to compensate for unforeseen system disturbances. We develop a decomposition method that partitions job operations into an ordered sequence of subsets. This decomposition identifies and resolves a \"crucial subset\" of scheduling decisions through the use of a branch-and-bound algorithm. We conduct computational experiments that demonstrate the performance of the approach under deterministic cases, and the robustness of the approach under a wide range of processing time perturbations. We show that the performance of the method is superior, particularly for low to medium levels of disturbances.