Facility location problems (FLPs) are a family of optimisation problems with significant social impact. This class of problems has been the subject of study since the 1960s, with classical approaches including the Weber problem and the p -Median problem. Currently, more complex variations of these problems are being investigated. In particular, the Induced p -Median Problem with Upgrades (IpMU) represents a variation of the classical p -Median problem, where the concepts of transport cost and time are separated as distinct metrics in the input graph of the problem. Furthermore, the problem includes a budget which allows one to relax the graph costs, reducing the cost of the edges, thus improving the associated routes between the designated medians and the customers. In this study, a metaheuristic algorithm, based on the Greedy Randomized Adaptive Search Procedure (GRASP), is proposed. A two-phase resolution scheme is defined, studying the median problem and the upgrading problem independently. In this approach, a larger set of state-of-the-art instances was analysed to ensure a fair comparison with previous proposals. In addition, the characteristics of the instances were studied to assess their complexity. The results obtained are promising when compared to the state-of-the-art, which is based entirely on mathematical programming models. The execution time was improved on average by two orders of magnitude for the harder instances, and the best known result was obtained in more than 99% of the tested instances.

Abstract Motivated by a real‐world vehicle routing application, we consider the maximum‐weight independent set problem: given a node‐weighted graph, find a set of independent (mutually nonadjacent) nodes whose node‐weight sum is maximum. Some of the graphs airsing in this application are large, having hundreds of thousands of nodes and hundreds of millions of edges. To solve instances of this size, we develop a new local search algorithm, which is a metaheuristic in the greedy randomized adaptive search framework. This algorithm, which we call METAMIS, uses a wider range of simple local search operations than previously described in the literature. We introduce data structures that make these operations efficient. A new variant of path‐relinking is introduced to escape local optima and so is a new alternating augmenting‐path local search move that improves algorithm performance. We compare an implementation of our algorithm with a state‐of‐the‐art openly available code on public benchmark sets, including some large instances with hundreds of millions of vertices. Our algorithm is, in general, competitive and outperforms this openly available code on large vehicle routing instances. We hope that our results will lead to even better MWIS algorithms.

This paper proposes an approach based on the Biased Random-Key Genetic Algorithm (BRKGA) metaheuristic to optimize hyperparameters of Convolutional Neural Network (CNN) architectures. We developed a Default version for computers that only have access to Central Processing Units (CPUs) and a version for Graphics Processing Units (GPUs) called cudaBRKGA-CNN, guaranteeing better computational performance during CNN training. Additionally, we developed a decoder that searches for CNN hyperparameters and returns competitive solutions. We compared the performance of our two proposed approaches with other state-of-the-art deep learning models using two datasets from the literature. Our approaches demonstrated high performance in terms of solution quality and convergence speed. Furthermore, our cudaBRKGA-CNN proposal presented competitive results and better computational time than the other evaluated models.

Motivated by a real-world vehicle routing application, we consider the maximum-weight independent set problem: Given a node-weighted graph, find a set of independent (mutually nonadjacent) nodes whose node-weight sum is maximum. Some of the graphs airsing in this application are large, having hundreds of thousands of nodes and hundreds of millions of edges. To solve instances of this size, we develop a new local search algorithm, which is a metaheuristic in the greedy randomized adaptive search (GRASP) framework. This algorithm, which we call METAMIS, uses a wider range of simple local search operations than previously described in the literature. We introduce data structures that make these operations efficient. A new variant of path-relinking is introduced to escape local optima and so is a new alternating augmenting-path local search move that improves algorithm performance. We compare an implementation of our algorithm with a state-of-the-art openly available code on public benchmark sets, including some large instances with hundreds of millions of vertices. Our algorithm is, in general, competitive and outperforms this openly available code on large vehicle routing instances. We hope that our results will lead to even better MWIS algorithms.

International audience

We investigate a variant of the many‐to‐many hub location‐routing problem which consists in partitioning the set of nodes of a graph into routes containing exactly one hub each, and determining an extra route interconnecting all hubs. A variable neighborhood descent with neighborhood structures based on remove/add, swap and exchange moves nested with routing and location operations is used as a local search procedure in a multistart algorithm. We also consider a sequential version of this local search in the multistart. In addition, a biased random‐key genetic algorithm working with a local search routine, which also considers routing and location operations, is applied to the problem. To compare the heuristic solutions, we develop an integer programming formulation which is solved with a branch‐and‐cut algorithm. Capacity and path elimination constraints are added in a cutting plane fashion. The separation algorithms are based on the computation of min‐cut trees and on the connected components of a support graph. Computational experiments were conducted on several benchmark instances of routing problems and show that the heuristics are effective on medium to large‐sized instances, while the branch‐and‐cut algorithm solves small to medium sized problems to optimality. These algorithms were also compared with a commercial hybrid solver showing that the heuristics are quite competitive. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(1), 54–90 2016

Cloud computing has recently emerged as a new technology for hosting and supplying services over the Internet. This technology has brought many benefits, such as eliminating the need for maintaining expensive computing hardware and allowing business owners to start from small and increase resources only when there is a rise in service demand. With an increasing demand for cloud computing, providing performance guarantees for applications that run over cloud become important. Applications can be abstracted into a set of virtual machines with certain guarantees depicting the quality of service of the application. In this paper, we consider the placement of these virtual machines across multiple data centers, meeting the quality of service requirements while minimizing the bandwidth cost of the data centers. This problem is a generalization of the NP-hard Generalized Quadratic Assignment Problem (GQAP). We formalize the problem and propose a novel algorithm based on a biased random-key genetic algorithm (BRKGA) to find near-optimal solutions for the problem. The experimental results show that the proposed algorithm is effective in quickly finding feasible solutions and it produces better results than a baseline aproach provided by a commercial solver and a multi-start algorithm.

In this paper we address the problem of picking a subset of bids in a general combinatorial auction so as to maximize the overall profit using the first-price model. This winner determination problem assumes that a single bidding round is held to determine both the winners and prices to be paid. We introduce six variants of biased random-key genetic algorithms for this problem. Three of them use a novel initialization technique that makes use of solutions of intermediate linear programming relaxations of an exact mixed integer linear programming model as initial chromosomes of the population. An experimental evaluation compares the effectiveness of the proposed algorithms with the standard mixed linear integer programming formulation, a specialized exact algorithm, and the best-performing heuristics proposed for this problem. The proposed algorithms are competitive and offer strong results, mainly for large-scale auctions.

In Overlapping Correlation Clustering (OCC), a number of objects are assigned to clusters. Two objects in the same cluster have correlated characteristics. As opposed to traditional clustering where objects are assigned to a single cluster, in OCC objects may be assigned to one or more clusters. In this paper, we present Biased Random-Key Genetic Algorithms for OCC. We present computational experiments such results outperformed the state of art methods for OCC.

Global optimization seeks a minimum or maximum of a multimodal function over a discrete or continuous domain. In this paper, we propose a biased random key genetic algorithm for finding approximate solutions for bound-constrained continuous global optimization problems subject to nonlinear constraints. Experimental results illustrate its effectiveness on some functions from CEC2006 benchmark (Liang et al. [2006]).

We introduce the $k$-Interconnected Multi-Depot Multi-Traveling Salesmen Problem, a new problem that resembles some network design and location routing problems but carries the inherent difficulty of not having a fixed set of depots or terminals. We propose a heuristic based on a biased random-key genetic algorithm to solve it. This heuristic uses local search procedures to best choose the terminal vertices and improve the tours of a given solution. We compare our heuristic with a multi-start procedure using the same local improvements and we show that the proposed algorithm is competitive.

This paper is a theoretical and experimental study of two related facility location problems that emanated from networking. Suppose we are given a network modeled as a directed graph G = (V, A), together with (not-necessarily-disjoint) subsets C and F of V, where C is a set of customer locations and F is a set of potential facility locations (and typically C ⊆ F). Our goal is to find a minimum sized subset F′ ⊆ F such that for every customer c ∊ C there are two locations f1, f2 ∊ F′ such that traffic from c to f1 and to f2 is routed on disjoint paths (usually shortest paths) under the network's routing protocols. Although we prove that this problem is impossible to approximate in the worst case even to within a factor of 2log1−εn for any ε > 0 (assuming no NP-complete language can be solved in quasipolynomial time), we show that the situation is much better in practice. We propose three algorithms that build solutions and determine lower bounds on the optimum solution, and evaluate them on several large real ISP topologies and on synthetic networks designed to reflect real-world LAN/WAN network structure. Our main algorithms are (1) an algorithm that performs multiple runs of a straightforward randomized greedy heuristic and returns the best result found, (2) a genetic algorithm that uses the greedy algorithm as a subroutine, and (3) a new “Double Hitting Set” algorithm. All three approaches perform surprising well, although, in practice, the most cost-effective approach is the multi-run greedy algorithm. This yields results that average within 0.7% of optimal for our synthetic instances and within 2.9% for our real-world instances, excluding the largest (and most realistic) one. For the latter instance, the other two algorithms come into their own, finding solutions that are more than three times better than those of the multi-start greedy approach. In terms of our motivating monitoring application, where every customer location can be a facility location, the results are even better. Here the above Double Hitting Set solution is 90% better than the default solution which places a monitor at each customer location - such comparisons help justify the proposed alternative monitoring scheme of [8]. Our results also show that, on average for our real-world instances, we could save an additional 18% by choosing the (shortest path) routes ourselves, rather than taking the simpler approach of relying on the network to choose them for us.

Abstract This article proposes a linear integer programming formulation and several heuristics based on GRASP and path relinking for the antibandwidth problem. In the antibandwidth problem, one is given an undirected graph with n nodes and must label the nodes in a way that each node receives a unique label from the set {1, 2,…, n }, such that, among all adjacent node pairs, the minimum difference between the node labels is maximized. Computational results show that only small instances of this problem can be solved exactly (to optimality) with a commercial integer programming solver and that the heuristics find high‐quality solutions in much less time than the commercial solver. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 58(3), 171–189 2011

Dynamic shortest-path algorithms update the shortest paths taking into account a change in an arc weight. This paper describes a new generic technique that allows the reduction of heap sizes used by several dynamic single-destination shortest-path algorithms. For unit weight changes, the updates can be done without heaps. These reductions almost always reduce the computational times for these algorithms. In computational testing, several dynamic shortest-path algorithms with and without the heap-reduction technique are compared. Speedups of up to a factor of 1.8 were observed using the heap-reduction technique on random weight changes and of over a factor of five on unit weight changes. We compare as well with Dijkstra's algorithm, which recomputes the paths from scratch. With respect to Dijkstra's algorithm, speedups of up to five orders of magnitude are observed.

Abstract Internet protocol (IP) traffic follows rules established by routing protocols. Shortest path‐based protocols, such as Open Shortest Path First (OSPF), direct traffic based on arc weights assigned by the network operator. Each router computes shortest paths and creates destination tables used for routing flow on the shortest paths. If a router has multiple outgoing links on shortest paths to a given destination, it splits traffic evenly over these links. It is also the role of the routing protocol to specify how the network should react to changes in the network topology, such as arc or router failures. In such situations, IP traffic is rerouted through the shortest paths not traversing the affected part of the network. This article addresses the issue of assigning OSPF weights and multiplicities to each arc, aiming to design efficient OSPF‐routed networks with minimum total weighted multiplicity (multiplicity multiplied by the arc length) needed to route the required demand and handle any single arc or router failure. The multiplicities are limited to a discrete set of values, and we assume that the topology is given. We propose an evolutionary algorithm for this problem, and present results applying it to several real‐world problem instances. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(1), 51–64 2007

The separation of intradomain and interdomain routing has been a key feature of the Internet's routing architecture from the early days of the ARPAnet. However, the appropriate "division of labor" between the two protocols becomes unclear when an Autonomous System (AS) has interdomain routes to a destination prefix through multiple border routers---a situation that is extremely common today because neighboring domains often connect in several locations. We believe that the current mechanism of early-exit or hot-potato routing---where each router in an AS directs traffic to the "closest" border router based on the intradomain path costs---is convoluted, restrictive, and sometimes quite disruptive. In this paper, we propose a flexible mechanism for routers to select the egress point for each destination prefix, allowing network administrators to satisfy diverse goals, such as traffic engineering and robustness to equipment failures. We present one example optimization problem that uses integer-programming techniques to tune our mechanism to improve network robustness. Experiments with topology and routing data from two backbone networks demonstrate that our solution is both simple (for the routers) and expressive (for the network administrators).

This paper proposes and tests variants of GRASP (greedy randomized adaptive search procedure) with path relinking for the three-index assignment problem (AP3). GRASP is a multistart metaheuristic for combinatorial optimization. It usually consists of a construction procedure based on a greedy randomized algorithm and of a local search. Path relinking is an intensification strategy that explores trajectories that connect high-quality solutions. Several variants of the heuristic are proposed and tested. Computational results show clearly that this GRASP for AP3 benefits from path relinking and that the variants considered in this paper compare well with previously proposed heuristics for this problem. GRASP with path relinking was able to improve the solution quality of heuristics proposed by Balas and Saltzman (1991), Burkard et al. (1996), and Crama and Spieksma (1992) on all instances proposed in those papers. We show that the random variable “time to target solution,” for all proposed GRASP with path-relinking variants, fits a two-parameter exponential distribution. To illustrate the consequence of this, one of the variants of GRASP with path relinking is shown to benefit from parallelization.

Abstract Intradomain traffic engineering aims to make more efficient use of network resources within an autonomous system. Interior Gateway Protocols such as OSPF (Open Shortest Path First) and IS‐IS (Intermediate System‐Intermediate System) are commonly used to select the paths along which traffic is routed within an autonomous system. These routing protocols direct traffic based on link weights assigned by the network operator. Each router in the autonomous system computes shortest paths and creates destination tables used to direct each packet to the next router on the path to its final destination. Given a set of traffic demands between origin‐destination pairs, the OSPF weight setting problem consists of determining weights to be assigned to the links so as to optimize a cost function, typically associated with a network congestion measure. In this article, we propose a genetic algorithm with a local improvement procedure for the OSPF weight‐setting problem. The local improvement procedure makes use of an efficient dynamic shortest path algorithm to recompute shortest paths after the modification of link weights. We test the algorithm on a set of real and synthetic test problems, and show that it produces near‐optimal solutions. We compare the hybrid algorithm with other algorithms for this problem illustrating its efficiency and robustness. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 36–56 2005

Abstract A frame relay service offers virtual private networks to customers by provisioning a set of long‐term private virtual circuits (PVCs) between customer endpoints on a large backbone network. During the provisioning of a PVC, routing decisions are made without any knowledge of future requests. Over time, these decisions can cause inefficiencies in the network and occasional offline rerouting of the PVCs is needed. In this paper, the offline PVC routing problem is formulated as an integer multicommodity flow problem with additional constraints and with an objective function that minimizes propagation delays and/or network congestion. We propose variants of a GRASP with path‐relinking heuristic for this problem. Experimental results for realistic‐size problems are reported, showing that the proposed heuristics are able to improve the solutions found with standard routing techniques. Moreover, the structure of our objective function provides a useful strategy for setting the appropriate value of its weight parameter, to achieve some quality of service (QoS) level defined by a desired balance between propagation delay and delay due to network congestion. © 2003 Wiley Periodicals, Inc.

Abstract Given an undirected graph with prizes associated with its nodes and weights associated with its edges, the prize‐collecting Steiner tree problem consists of finding a subtree of this graph which minimizes the sum of the weights of its edges plus the prizes of the nodes not spanned. In this paper, we describe a multistart local search algorithm for the prize‐collecting Steiner tree problem, based on the generation of initial solutions by a primal‐dual algorithm using perturbed node prizes. Path‐relinking is used to improve the solutions found by local search and variable neighborhood search is used as a post‐optimization procedure. Computational experiments involving different algorithm variants are reported. Our results show that the local search with perturbations approach found optimal solutions on nearly all of the instances tested. © 2001 John Wiley & Sons, Inc.

We propose FORTRAN subroutines for approximately solving the feedback vertex and arc set problems on directed graphs using a Greedy Randomized Adaptive Search Procedure (GRASP). Implementation and usage of the package is outlined and computational experiments are reported illustrating solution quality as a function of running time.

In this paper, we introduce the truncated primal-infeasible dual-feasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum-cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code PDNET is tested on a large set of standard minimum-cost network flow test problems. Computational results indicate that the implementation is competitive with state-of-the-art network flow codes. © 2000 John Wiley & Sons, Inc.

We describe Fortran subroutines for finding approximate solutions of the maximum planar subgraph problem (graph planarization) using a Greedy Randomized Adaptive Search Procedure (GRASP). The design and implementation of the code are described in detail. Computational results with the subroutines illustrate the quality of solutions found as a function of number of GRASP iterations.

Let G=(V, E) be an undirected graph where V and E are the sets of vertices and edges of G, respectively. A subset of the vertices S ⊆ V is independent if all of its members are pairwise nonadjacent, i.e., have no edge between them. A solution to the NP-hard maximum independent set problem is an independent set of maximum cardinality. This article describes gmis, a set of Fortran subroutines to find an approximate solution of a maximum independent set problem. A greedy randomized adaptive search procedure (GRASP) is used to produce the solutions. The algorithm is described in detail. Implementation and usage of the package is outlined, and computational experiments are reported, illustrating solution quality as a function of running time.

A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. In this paper, we describe a GRASP for the graph planarization problem, extending the heuristic of Goldschmidt and Takvorian [Networks 24 (1994) 69–73]. We review the basic concepts of GRASP: construction and local search algorithms. The implementation of GRASP for graph planarization is described in detail. Computational experience on a large set of standard test problems is presented. On almost all test problems considered, the new heuristic either matches or finds a better solution than previously described graph planarization heuristics. In several cases, previously unknown optima solutions are found. © 1997 John Wiley & Sons, Inc. Networks 29: 173–189, 1997

We describe Fortran subroutines for finding approximate solutions of sparse instances of the Quadratic Assignment Problem (QAP) using a Greedy Randomized Adaptive Search Procedure (GRASP). The design and implementation of the code are described in detail. Computational results comparing the new subroutines with a dense version of the code (Algorithm 754, ACM TOMS) show that the speedup increases with the sparsity of the data.

In the NP-complete quadratic assignment problem (QAP), n facilities are to be assigned to n sites at minimum cost. The contribution of assigning facility i to site k and facility j to site l to the total cost is f ij d kl , where f ij is the flow between facilities i and j , and d kl is the distance between sites k and l . Only very small ( n ≤20) instances of the QAP have been solved exactly, and heuristics are therefore used to produce approximate solutions. This article describes a set of Fortran subroutines to find approximate solutions to dense quadratic assignment problems, having at least one symmetric flow or distance matrix. A greedy, randomized, adaptive search procedure (GRASP) is used to produce the solutions. The design and implementation of the code are described in detail, and extensive computational experiments are reported, illustrating solution quality as a function of running time.

This paper describes data structures and programming techniques used in an implementation of Karmarkar's algorithm for linear programming. Most of our discussion focuses on applying Gaussian elimination toward the solution of a sequence of sparse symmetric positive definite systems of linear equations, the main requirement in Karmarkar's algorithm. Our approach relies on a direct factorization scheme, with an extensive symbolic factorization step performed in a preparatory stage of the linear programming algorithm. An interpretative version of Gaussian elimination makes use of the symbolic information to perform the actual numerical computations at each iteration of algorithm. We also discuss ordering algorithms that attempt to reduce the amount of fill-in in the LU factors, a procedure to build the linear system solved at each iteration, the use of a dense window data structure in the Gaussian elimination method, a preprocessing procedure designed to increase the sparsity of the linear programming coefficient matrix, and the special treatment of dense columns in the coefficient matrix. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.