In this article, we address a real life optimization problem, the rail track inspection scheduling problem. This problem consists of scheduling railway network inspection tasks. The objective is to minimize the total deadhead distance while performing all inspection tasks. Different 0–1 integer formulations for the problem are presented. A heuristic based on both Benders and Dantzig-Wolfe decompositions is proposed to solve this rich arc routing problem. Its performance is analyzed on a real life dataset provided by the French national railway company. The proposed algorithm is compared to a dynamic programming-based heuristic. Its ability to schedule the inspection tasks of 1 year on a sparse graph with thousand nodes and arcs is assessed. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>We reconsider the successful approaches that we adopted in the past to solve Train Timetabling, Train Platforming, Train-Unit Assignment, and Crew Assignment problems arising in railway planning. We try to unify these approaches under a common framework, noting that they are all formulated as variants of a fairly general version of integer multicommodity flow, and discussing the solution methods and modelling issues that we found most relevant. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>Consider a directed network in which each arc can fail with some specified probability. An entity arrives on this network at a designated origin node and traverses the network in a random-walk fashion until it either terminates at a destination node, or until an arc fails while being traversed. We study the problem of assessing the probability that the random walk reaches the destination node, which we call the survival probability of the network. Complicating our analysis is the assumption that certain arcs have “memory,” in the sense that after a memory arc is successfully traversed, it cannot fail on any subsequent traversal during the walk. We prove that this problem is #P-hard, provide methods for obtaining lower and upper bounds on the survival probability, and demonstrate the effectiveness of our bounding methods on randomly generated networks. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>In this article, we investigate the stochastic maximum weight forest problem. We present two mathematical formulations for the problem: a polynomial sized one based on the characterization of forests in graphs and a formulation with an exponential number of constraints. We give a proof of the correctness of the new formulation and present a polynomial reduction from the set cover problem to give some insight about the complexity of this problem. We introduce an L-shaped decomposition approach for the polynomial formulation, thus allowing the optimal solution of large scale instances with up to 90 nodes. Finally, we propose a Kruskal based variable neighborhood search (VNS) metaheuristic to compute near optimal solutions with significantly less computational effort. Our numerical results show that the VNS approach provides tight near optimal solutions with a gap less than 1% for most of the instances. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>The reload cost spanning tree problem (RCSTP) is an NP-hard problem, where we are given a set of nonnegative pairwise demands between nodes, each edge is colored and a reload cost is incurred when a color change occurs on the path between a pair of demand nodes. The goal is to find a spanning tree with minimum total reload cost. We propose a tree–nontree edge swap neighborhood for the RCSTP and an efficient way to search this neighborhood using preprocessed information. We then embed this edge swap neighborhood within a local search and a tabu search heuristic. We also discuss an initial solution procedure that is used by the local search and tabu search heuristic in a multistart framework. On a test set of 630 instances (that includes benchmark instances from Gamvros et al. [6]), the local search solution improves upon the initial solution in 416 instances by an average of 23.62%, and the tabu search solution improves upon the local search solution in 364 instances by an average of 35.79%. Out of 495 test instances from this set that we know the optimal solutions for, the initial solution is optimal 113 times, the local search solution is optimal 224 times, and the tabu search solution is optimal 481 times. On a second set of benchmark instances from Khalil and Singh [9], the tabu search solution improves upon the best known solution in 32 out of 44 instances. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>We consider the impact of scheduling disciplines on performance of routing in the framework of adversarial queuing. We propose an adversarial model which reflects stalling of packets due to transient failures and explicitly incorporates feedback produced by a network when packets are stalled. This adversarial model provides a methodology to study stability of routing protocols when flow-control and congestion-control mechanisms affect the volume of traffic. We show that any scheduling policy that is universally stable, in the regular model of routing that additionally allows packets to have two priorities, remains stable in the proposed adversarial model. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>Given a connected and undirected graph *G*, the degree preserving spanning tree problem (DPSTP) asks for a spanning tree of *G* with the maximum number of vertices having the same degree in the tree and in *G*. These are called full degree vertices. We introduce integer programming formulations, valid inequalities and four exact solution approaches based on different formulations. Two branch-and-bound procedures, a branch-and-cut (BC) algorithm and an iterative probing combinatorial Benders decomposition method are introduced here. The problem of optimally lifting one of the classes of valid inequalities proposed here is equivalent to solving a DPSTP instance, for a conveniently defined subgraph of *G*. We thus apply one of the proposed methods to optimally lift these cuts, within the other solution methods. In doing so, two additional algorithms, a hybrid Benders decomposition and a hybrid BC are proposed. Extensive computational experiments are conducted with the solution algorithms introduced in this study. © 2014 Wiley Periodicals, Inc. NETWORKS, 2015

In the freight car dispatching problem, empty freight cars have to be assigned to known demands respecting a given time horizon and certain constraints. The goal is to minimize the resulting transportation costs. One of the constraints is that customers can specify the type of cars they want. It is possible, however, that cars of certain types can be substituted by other cars, either in a 1-to-1 fashion or at different exchange rates. We show that these substitutions make the dispatching problem hard to solve and hard to approximate. We model the dispatching problem as an integral generalized transportation problem on a bipartite graph. Using rounding techniques, the LP-relaxation can be transformed to a transportation schedule violating some of the constraints slightly. Under an additional assumption on the cost function, we fix this violation and derive a 4-approximation of the problem. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>We introduce the class of spot-checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero-sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium (SSE) is more relevant for this problem and we give a mixed integer programming (MIP) formulation for this problem. We show that the computation of such an equilibrium is NP-hard. More generally, we prove that it is NP-hard to compute a SSE in a polymatrix game, even if the game is pairwise zero-sum. Then, we give some bounds on the *price of spite*, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

We study the incremental facility location problem, wherein we are given an instance of the uncapacitated facility location problem (UFLP) and seek an incremental sequence of opening facilities and an incremental sequence of serving customers along with their fixed assignments to facilities open in the partial sequence. We say that a sequence has a competitive ratio of *k*, if the cost of serving the first *ℓ* customers in the sequence is at most *k* times the optimal solution for serving any *ℓ* customers for all possible values of *ℓ*. We provide an incremental framework that computes a sequence with a competitive ratio of at most eight and a worst-case instance that provides a lower bound of three for any incremental sequence. We also present the results of our computational experiments carried out on a set of benchmark instances for the UFLP. The problem has applications in multistage network planning. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

The quadratic minimum spanning tree problem (QMSTP) consists of finding a spanning tree of a graph *G* such that a quadratic cost function is minimized. In its adjacent only version (AQMSTP), interaction costs only apply for edges that share an endpoint. Motivated by the weak lower bounds provided by formulations in the literature, we present a new linear integer programming formulation for AQMSTP. In addition to decision variables assigned to the edges, it also makes use of variables assigned to the stars of *G*. In doing so, the model is naturally linear (integer), without the need of implementing usual linearization steps, and its linear programming relaxation better estimates the interaction costs between edges. We also study a reformulation derived from the new model, obtained by projecting out the decision variables associated with the stars. Two exact solution approaches are presented: a branch-and-cut-and-price algorithm, based on the first formulation, and a branch-and-cut algorithm, based on its projection. Our computational results indicate that the lower bounds introduced here are much stronger than previous bounds in the literature. Being designed for the adjacent only case, our duality gaps are one order of magnitude smaller than the Gilmore–Lawler lower bounds for AQMSTP. As a result, the two exact algorithms introduced here outperform the previous exact solution approaches in the literature. In particular, the branch-and-cut method we propose managed to solve AQMSTP instances with as many as 50 vertices to proven optimality. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

In this article, we study the (*k,c*)-coloring problem, a generalization of the vertex coloring problem where we have to assign *k* colors to each vertex of an undirected graph, and two adjacent vertices can share at most *c* colors. We propose a new formulation for the (*k,c*)-coloring problem and develop a Branch-and-Price algorithm. We tested the algorithm on instances having from 20 to 80 vertices and different combinations for *k* and *c*, and compare it with a recent algorithm proposed in the literature. Computational results show that the overall approach is effective and has very good performance on instances where the previous algorithm fails. © 2014 Wiley Periodicals, Inc. NETWORKS, 2014

Given an edge-weighted graph *G* and two distinct vertices *s* and *t* of *G*, the next-to-shortest path problem asks for a path from *s* to *t* of minimum length among all paths from *s* to *t* except the shortest ones. In this article, we consider the version where *G* is directed and all edge weights are positive. Some properties of the requested path are derived when *G* is an arbitrary digraph. In addition, if *G* is planar, an -time algorithm is proposed, where *n* is the number of vertices of *G*. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 205–211 2015

The aircraft scheduling problem (ASP) is the real-time problem of scheduling takeoff and landing operations at a congested airport in a given time horizon, taking into account the runways and the air segments in the terminal maneuvering area . The ASP can be viewed as a job shop scheduling problem with additional real-world constraints. Compared with the current literature based on job shop scheduling applied to solve the ASP, we enrich the existing models by including new formulations of relevant practical constraints. We introduce and analyze three alternative ASP formulations, in which the objective function is the minimization of delay propagation with respect to the off-line timetable. Scheduling rules, heuristic and exact methods are implemented and tested on instances from the Roma Fiumicino airport, in Italy. Computational experiments show that practical-size instances are solved to near-optimality by our branch and bound algorithm in a few seconds of computation. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 212–227 2015

]]>To model and solve optimization problems arising in public transportation, data about the passengers are necessary and have to be included in the models in any phase of the planning process. Many approaches assume a two-step procedure: in a first step, the data about the passengers are distributed over the public transportation network (PTN) using traffic-assignment procedures. In a second step, the actual planning of lines, timetables, and so forth takes place. This approach ignores that, assuming that the network is sufficiently dense, for most passengers, there are many possible ways to reach their destinations in the PTN, thus the actual connections the passengers will take strongly depend on the decisions made during the planning phase. In this article, we investigate the influence of integrating the traffic assignment procedure in the optimization process on the complexity of the line planning problem. Our objective is to maximize the passengers' benefit, namely to minimize the overall travel time of the passengers in the network. We present new models and systematically analyze their complexities. Exploiting a relation to the resource-constrained shortest path problem, we are able to derive pseudopolynomial and polynomial algorithms for special cases. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 228–243 2015

]]>Traditionally, the coordination of multiple traffic signals and the traffic assignment problem in an urban street network are considered as two separate optimization problems. However, it is easy to see that the traffic assignment has an influence on the optimal signal coordination and, vice versa, a change in the signal coordination changes the optimal traffic assignment. In this article, we present a cyclically time-expanded network and a corresponding mixed integer linear programming formulation for simultaneously optimizing both the coordination of traffic signals and the traffic assignment in an urban street network. Although the new cyclically time-expanded network provides a model of both traffic and signals close to reality, it still has the advantage of a linear objective function. Using this model, we compute optimized signal coordinations and traffic assignment on real-world street networks. To evaluate the practical relevance of the computed solutions, we conduct extensive simulation experiments using two established traffic simulation tools that reveal the advantages of our model. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 244–261 2015

]]>Reducing traffic congestion via toll pricing has been a central topic in the operations research and transportation literature and, recently, it has been implemented in several cities all over the world. Since, in practice, it is not feasible to impose tolls on every edge of a given traffic network, we study the resulting mathematical problem of computing tolls on a predefined subset of edges of the network so as to minimize the total travel time of the induced equilibrium flow. We first present an analytical study for the special case of parallel edge networks highlighting the intrinsic complexity and nonconvexity of the resulting optimization problem. We then present algorithms for general networks for which we systematically test the solution quality for large-scale network instances. Finally, we discuss the related optimization problem of computing tolls subject to a cardinality constraint on the number of edges that have tolls. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 262–285 2015

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