In the so-called network pricing problem an authority owns some arcs of the network and tolls them, while users travel between their origin and destination choosing their minimum cost path. In this article, we consider a unit toll scheme, and in particular the cases where the authority is imposing either the same toll on all of its arcs, or a toll proportional to a given parameter particular to each arc (for instance a per kilometer toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial, proposing ad-hoc solution algorithms and relating these problems to the parametric shortest path problem. We then address a robust approach using an interval representation to take into consideration uncertainty on parameters. We show how to modify the algorithms for the deterministic case to solve the robust counterparts, maintaining their complexity class. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

]]>We model and solve the Rainbow Cycle Cover Problem (RCCP). Given a connected and undirected graph and a coloring function that assigns a color to each edge of from the finite color set , a cycle whose edges have all different colors is called a rainbow cycle. The RCCP consists of finding the minimum number of disjoint rainbow cycles covering . The RCCP on general graphs is known to be NP-complete. We model the RCCP as an integer linear program, we derive valid inequalities and we solve it by branch-and-cut. Computational results are reported on randomly generated instances. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

]]>We consider the problem of optimally locating a single facility anywhere in a network to serve both on-network and off-network demands. Off-network demands occur in a Euclidean plane, while on-network demands are restricted to a network embedded in the plane. On-network demand points are serviced using shortest-path distances through links of the network (e.g., on-road travel), whereas demand points located in the plane are serviced using more expensive Euclidean distances. Our base objective minimizes the total weighted distance to all demand points. We develop several extensions to our base model, including: (i) a threshold distance model where if network distance exceeds a given threshold, then service is always provided using Euclidean distance, and (ii) a minimax model that minimizes worst-case distance. We solve our formulations using the “Big Segment Small Segment” global optimization method, in conjunction with bounds tailored for each problem class. Computational experiments demonstrate the effectiveness of our solution procedures. Solution times are very fast (often under one second), making our approach a good candidate for embedding within existing heuristics that solve multi-facility problems by solving a sequence of single-facility problems. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

]]>Designing networks with specified collective properties is useful in a variety of application areas, enabling the study of how given properties affect the behavior of network models, the downscaling of empirical networks to workable sizes, and the analysis of network temporal evolution. Despite the importance of the task, there currently exists a gap in our ability to systematically generate networks that adhere to theoretical guarantees for the given property specifications. In thisarticle, we propose the use of Mixed-Integer Linear Optimization modeling and solution methodologies to address this *Network Generation Problem*. We present useful modeling techniques and apply them to mathematically express and constrain a broad class of network properties in the context of an optimization formulation. We derive complete formulations for the generation of networks that attain specified levels of connectivity, spread, assortativity and robustness, and we illustrate these via a number of computational case studies. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity was not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected -free-minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so-called semi-minimal disconnected cut is still polynomial-time solvable on planar graphs. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

]]>In this paper, we consider the Mixed Capacitated General Routing Problem which is a combination of the Capacitated Vehicle Routing Problem and the Capacitated Arc Routing Problem. The problem is also known as the Node, Edge, and Arc Routing Problem. We propose a Branch-and-Cut-and-Price algorithm for obtaining optimal solutions to the problem and present computational results based on a set of standard benchmark instances. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 161–184 2016

]]>This paper deals with the problem of capacity expansion of a network under independent uncertain demands defined by interval sets. In this context, decisions about capacity expansion must be made before the demands are revealed. Standard robust models require the definition of an uncertainty domain and look for the minimum cost solution able to satisfy any demand within this domain. We propose, justify, and illustrate an alternative robust model based on a bi-objective formulation. Therefore, in addition to the cost criterion, we consider a second criterion, related to the Quality of Service, which measures the ability of a solution to handle any demand. The decision-maker can be interested in efficient solutions offering a compromise between these criteria. We study the complexity of the enumeration of the corresponding nondominated set, and propose exact and approximation algorithms. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 185–199 2016

]]>We propose new robust models for handling right hand side uncertainty in linear problems. A cutting plane-like method could be devised to solve the resulting robust problems, but the subproblem to solve at each step involves a bilinear objective. Upper approximations are thus constructed based on linear decision and zero-order rules on the adjustable variables. Tractable reformulations are given on some uncertainty sets arising in practice. Heuristics are also proposed to compute lower bounds. To assess the methodology we consider its application to the capacity assignment problem. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 200–211 2016

]]>We introduce the driver assignment vehicle routing problem, DAVRP. In this problem, drivers are assigned to customers before demand is known, and after demand is known a routing schedule has to be made such that every driver visits at least a fraction *α* of its assigned customers. We present a solution procedure to investigate how much transportation costs increase by adhering to the driver assignments. Furthermore, we distinguish between the case in which customers that are not visited by their assigned driver are visited by backup drivers only, and the case in which slack capacity of regular drivers is utilized to visit these customers. We use randomly generated instances of the DAVRP to provide examples where the difference in transportation costs is substantial. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 212–223 2016

Availability, defined as the fraction of time a network service is operative, is a key network service parameter. Dedicated protection increases availability but also the cost. Shared protection instead decreases the cost, but also the availability. In this article, we formulate and solve an integer linear programming (ILP) model for the problem of minimizing the backup resources required by a shared-protected static optical network whilst guaranteeing an availability target per connection. The main research challenge is dealing with the nonlinear expression for the availability constraint. Taking the working/backup routes and the availability requirements as input data, the ILP model identifies the set of connections sharing backup resources in any given network link. We also propose a greedy heuristic to solve large instances in much shorter time than the ILP model with low levels of relative error (2.49% average error in the instances studied) and modify the ILP model to evaluate the impact of wavelength conversion. Results show that considering availability requirements can lead up to 56.4% higher backup resource requirements than not considering them at all, highlighting the importance of availability requirements in budget estimation. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 224–237 2016

]]>Given a graph *G* whose edges are perfectly reliable and whose nodes each operate independently with probability the *node reliability* of *G* is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce; it is the analogous node measure of robustness to the well studied *all-terminal reliability*, where the nodes are perfectly reliable but the edges fail randomly. In sharp contrast to what is known about the roots of the all-terminal reliability polynomial, we show that the node reliability polynomial of any connected graph on at least three nodes has a nonreal polynomial root, the collection of real roots of all node reliability polynomials is unbounded, and the collection of complex roots of all node reliability polynomials is dense in the entire complex plane. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 238–246 2016