Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets. The objective is to minimize the sum of the distances of the chosen set of edges for the worst positions of the vertices in their uncertainty sets. We first prove that these problems are [Formula: see text]-hard even when the feasible sets consist either of all spanning trees or of all s – t paths. Given this hardness, we propose an exact solution algorithm combining integer programming formulations with a cutting plane algorithm, identifying the cases where the separation problem can be solved efficiently. We also propose a conservative approximation and show its equivalence to the affine decision rule approximation in the context of Euclidean distances. We compare our algorithms to three deterministic reformulations on instances inspired by the scientific literature for the Steiner tree problem and a facility location problem. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1276 .

Due to large-scale applications and services, cloud computing infrastructures are experiencing an ever-increasing demand for computing resources. At the same time, the overall power consumption of data centers has been rising beyond 1% of worldwide electricity consumption. The usage of renewable energy in data centers contributes to decreasing their carbon footprint and overall electricity costs. Several green-energy-aware resource allocation approaches have been studied recently. None of them takes advantage of the joint migration ofjobsandenergyin green data centers to increase energy efficiency. This paper presents an optimization approach for energy-efficient resource allocation in mini data centers. The observed momentum around edge computing makes the design of geographically distributed mini data centers highly desirable. Our solution exploits both virtual machines (VMs) and energy migrations between green compute nodes in mini data centers. These nodes have energy harvesting, storage, and transport capabilities. They enable the migration of VMs and energy across different nodes. Compared to VM allocation alone, joint-optimization of VM and energy allocation reduces utility electricity consumption by up to 22%. This reduction can reach up to 28.5% for the same system when integrating less energy-efficient servers. The gains are demonstrated using simulation and a Mixed Integer Linear Programming formulation for the resource allocation problem. Furthermore, we show how our solution contributes to sustaining the energy consumption of old-generation and less efficient servers in mini data centers.

In this paper, we consider a variant of adaptive robust combinatorial optimization problems where the decision maker can prepare K solutions and choose the best among them upon knowledge of the true data realizations. We suppose that the uncertainty may affect the objective and the constraints through functions that are not necessarily linear. We propose a new exact algorithm for solving these problems when the feasible set of the nominal optimization problem does not contain too many good solutions. Our algorithm enumerates these good solutions, generates dynamically a set of scenarios from the uncertainty set, and assigns the solutions to the generated scenarios using a vertex p-center formulation, solved by a binary search algorithm. Our numerical results on adaptive shortest path and knapsack with conflicts problems show that our algorithm compares favorably with the methods proposed in the literature. We additionally propose a heuristic extension of our method to handle problems where it is prohibitive to enumerate all good solutions. This heuristic is shown to provide good solutions within a reasonable solution time limit on the adaptive knapsack with conflicts problem. Finally, we illustrate how our approach handles nonlinear functions on an all-or-nothing subset problem taken from the literature. Summary of Contribution: Our paper describes a new exact algorithm for solving adaptive robust combinatorial optimization problems when the feasible set of the nominal optimization problems does not contain too many good solutions. Its development relies on a progressive relaxation of the problem augmented with a row-and-column generation technique. Its efficient execution requires a reformulation of this progressive relaxation, coupled with dominance rules and a binary search algorithm. The proposed algorithm is amenable to exploiting the special structures of the problems considered as illustrated with various applications throughout the paper. A practical view is provided by the proposition of a heuristic variant. Our computational experiments show that our proposed exact solution method outperforms the existing methodologies and therefore pushes the computational envelope for the class of problems considered.

We consider robust variants of the bin packing problem with uncertain item sizes. Specifically we consider two uncertainty sets previously studied in the literature. The first is budgeted uncertainty (the $U^\Gamma$ model), in which at most $\Gamma$ items deviate, each reaching its peak value, while other items assume their nominal values. The second uncertainty set, the $U^\Omega$ model, bounds the total amount of deviation in each scenario. We show that a variant of the Next-cover algorithm is a $2$ approximation for the $U^\Omega$ model, and another variant of this algorithm is a $2\Gamma$ approximation for the $U^\Gamma$ model. Unlike the classical bin packing problem, it is shown that (unless $\mathcal{P}=\mathcal{NP}$) no asymptotic approximation scheme exists for the $U^\Gamma$ model, for $\Gamma=1$. This motivates the question of the existence of a constant approximation factor algorithm for the $U^\Gamma$ model. Our main result is to answer this question by proving a (polynomial-time) $4.5$ approximation algorithm, based on a dynamic-programming approach.

Abstract We consider the context of a telecommunication company that is at the same time an infrastructure operator and a service provider. When planning its network expansion, the company can leverage over its knowledge of the subscriber dynamic to better optimize the network dimensioning, therefore avoiding unnecessary costs. In this work, the network expansion represents the deployment and/or reinforcement of several technologies (e.g., 2G, 3G, 4G), assuming that subscribers to a given technology can be served by this technology or older ones. The operator can influence subscriber dynamic by subsidies. The planning is made over a discretized time horizon while some strategic guideline requirements are required at the end of the time horizon. Following classical models, we consider that the willingness of customers for shifting to a new technology follows an S‐shape piecewise constant function. We propose a mixed‐integer linear programming formulation, improved through several valid inequalities and a heuristic algorithm. We assess the formulation numerically on real instances.

Abstract We consider a time‐dependent shortest path problem with possible waiting at some nodes of the graph and a global bound W on the total waiting time. The goal is to minimize the time traveled along the edges of the path, not including the waiting time. We prove that the problem can be solved in polynomial time when the travel time functions are piecewise linear and continuous. The algorithm relies on a recurrence relation characterized by a bound ω on the total waiting time, where 0 ≤ ω ≤ W. We show that only a small number of values ω1, ω2, …, ωK need to be considered, where K depends on the total number of breakpoints of all travel time functions.

We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min–max–min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Γ of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min–max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min–max–min problem is NP-hard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on min-knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times.

Abstract Flow thinning (FT) is a traffic protection mechanism for communication networks with variable link capacities, for example wireless networks. With FT, end‐to‐end traffic demands use dedicated logical tunnels, for example MPLS tunnels, whose nominal capacity is subject to thinning in order to follow fluctuations in link capacities availability. Moreover, instantaneous traffic of each demand is throttled at its originating node accordingly to the current total capacity available on the demand's dedicated tunnels so that the network is always capable of carrying the admitted traffic. In this paper, we deal with efficient, implementable versions of FT, referred to as affine FT (AFT) and quadratic FT (QFT). By deriving appropriate link availability state and path generation algorithms, we show how real‐life network dimensioning problems for AFT/QFT can be efficiently treated using a proper characterization of the network link availability states. Results of a numerical study illustrate tractability of the cost minimization problems, and assess efficiency of AFT/QFT as compared with other protection mechanisms.

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Abstract We study a variant of the shortest path problem in a congested environment. In this setting, the travel time of each arc is represented by a piecewise continuous affine function of departure time. Besides, the driver is allowed to wait at nodes to avoid wasting time in traffic. While waiting, the driver is able to perform useful tasks for her job or herself, so the objective is to minimize only driving time. Although optimal solutions may contain cycles and pseudo‐polynomially many arcs, we provide a representation of the solutions that is polynomial in the absolute value of the inverse of the slopes as well as in the dimensions of the graph. We further prove that the problem is ‐Hard when the slopes are integer. We introduce a restriction of the problem where waits must be integer and propose pseudo‐polynomial algorithms for the latter. We also provide a pseudo‐FPTAS, polynomial in the ratio between the bound on the total waiting time and the minimum travel time. Finally, we discuss harder variants of the problem and show their inapproximability.

We experiment with an alternative routing scheme for the robust network loading problem with demand uncertainty. Named k‐adaptive, it is based on the fact that the decision‐maker chooses k second‐stage solutions and then commits to one of them only after realization of the uncertainty. This routing scheme, with its corresponding k‐partition of the uncertainty set, is dynamically defined under an iterative method to sequentially improve the solution. The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control time performance. We compare our k‐adaptive results with the ones obtained through other routing schemes and also verify the effectiveness of the methods utilized using several realistic networks from SNDlib and other sources.

The expansion of a telecommunications network faces two sources of uncertainty, which are the demand for traffic that will transit through the expanded network and the outsourcing cost that the network operator will have to pay to handle the traffic that exceeds the capacity of her network. The latter is determined by the future cost of telecommunications services, whose negative correlation with the total demand is empirically measured in the literature through the price elasticity of demand.

The present study deals with Elastic Flow Rerouting (EFR)—an original traffic restoration strategy for protecting traffic flows in communication networks (including wireless networks) against multiple link failures. EFR aims at alleviating the trade‐off between practicability of traffic restoration and the cost of network resources observed in existing networking solutions. We present an extension of EFR capable of managing multiple partial link failures. We describe EFR and its extension, formulate the EFR related optimization problems, and discuss approaches for their resolution. We also discuss numerical results illustrating effectiveness of EFR in terms of the link capacity cost. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(4), 267–281 2015

We study the robust constrained shortest path problem under resource uncertainty. After proving that the problem is in the strong sense for arbitrary uncertainty sets, we focus on budgeted uncertainty sets introduced by Bertsimas and Sim (2003) and their extension to variable uncertainty by Poss (2013). We apply classical techniques to show that the problem with capacity constraints can be solved in pseudopolynomial time. However, we prove that the problem with time windows is in the strong sense when is not fixed, using a reduction from the independent set problem. We introduce then new robust labels that yield dynamic programming algorithms for the problems with time windows and capacity constraints. The running times of these algorithms are pseudopolynomial when is fixed, exponential otherwise. We present numerical results for the problem with time windows which show the effectiveness of the label-setting algorithm based on the new robust labels. Our numerical results also highlight the reduction in price of robustness obtained when using variable budgeted uncertainty instead of classical budgeted uncertainty. © 2015 Wiley Periodicals, Inc.NETWORKS, Vol. 66(2), 98–111 2015

Leveraging the concept of controlled node mobility in this paper we develop an algorithm for tracking and controlling proximity malware propagation in Wireless Sensor Networks (WSN). Our proposal aims at: (i) notifying the nodes of malwarepropagation, (ii) leading a flying robot along a path in the WSNthat guarantees the minimum recovery time to (iii) heal the infected nodes. We formulate the targeted curing problem as a binary integer problem and determine the optimal solution by a central solver. We use the analytical result as a benchmark to evaluate the recovery time of the proposed solution. The achieved results show a satisfactory performance in terms of tracking the presence of an ongoing epidemic and healing the nodes.

Given a graph with nonnegative edge weights and node pairs Q, we study the problem of constructing a minimum weight set of edges so that the induced subgraph contains at least K edge-disjoint paths containing at most L edges between each pair in Q. Using the layered representation introduced by Gouveia [Gouveia, L. 1998. Using variable redefinition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS J. Comput. 10(2) 180–188], we present a formulation for the problem valid for any K, L ≥ 1. We use a Benders decomposition method to efficiently handle the large number of variables and constraints. We show that our Benders cuts contain constraints used in previous studies to formulate the problem for L = 2, 3, 4, as well as new inequalities when L ≥ 5. Whereas some recent works on Benders decomposition study the impact of the normalization constraint in the dual subproblem, we focus here on when to generate the Benders cuts. We present a thorough computational study of various branch-and-cut algorithms on a large set of instances including the real-based instances from SNDlib. Our best branch-and-cut algorithm combined with an efficient heuristic is able to solve the instances significantly faster than CPLEX 12 on the extended formulation.

Abstract Affinely Adjustable Robust Counterparts provide tractable alternatives to (two‐stage) robust programs with arbitrary recourse. Following Ouorou and Vial, we apply them to robust network design with polyhedral demand uncertainty, introducing the notion of affine routing. We compare the new affine routing scheme to the well‐studied static and dynamic routing schemes for robust network design. It is shown that affine routing can be seen as a generalization of the widely used static routing while still being tractable and providing cheaper solutions. We investigate properties of the demand polytope under which affine routings reduce to static routings and also develop conditions on the uncertainty set leading to dynamic routings being affine. We show however that affine routings suffer from the drawback that (even totally) dominated demand vectors are not necessarily supported by affine solutions. Uncertainty sets have to be designed accordingly. Finally, we present computational results on networks from SNDlib. We conclude that for these instances the optimal solutions based on affine routings tend to be as cheap as optimal network designs for dynamic routings. In this respect the affine routing principle can be used to approximate the cost for two‐stage solutions with free recourse which are hard to compute. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013