We consider a time-dependent network with a continuous-time variable, in which time constraints are imposed both on the arrival times and on the waiting times at the nodes. Under certain continuity assumptions, we prove the existence of optimal paths, and we show that the optimal value function is lower-semicontinuous. We present an exact solution algorithm, which computes both the optimal value function and the corresponding optimal paths. This algorithm is based on a Dijkstra-like interpretation of a decreasing order of time algorithm, which allows the generalization of this method to a heuristic search algorithm. Moreover, we present an approximation procedure for the computation of the optimal value function and the corresponding optimal paths in a time-dependent first-in first-out (FIFO) network. This method allows for the iterative construction of paths of monotone decreasing cost, starting from a path that is computable in polynomial time. We prove the correctness and termination of both algorithms. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

The authors analyze two-person zero-sum search games of the following type. Play takes place on a network; the hider must choose a node and remain there while the searcher can choose the node at which he starts. To detect the hider, the searcher needs to conduct a search at the node chosen by the hider. Searching a node involves a cost which can vary from node to node. In addition to the search costs, the searcher also incurs travelling costs represented by distances on the edges. The costs are known to both players and the searcher wants to minimize his total costs. An upper bound for the value of the game is obtained and a lower bound when the network has all its edge lengths the same. Restricting attention to networks which have all their edges of the same length, the upper, and lower bounds are shown to coincide for some networks including Hamiltonian ones. Some results for the star and line networks are also given. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

We consider the problem of covering the edges of a graph by a sequence of matchings subject to the constraint that each edge e appears in at least a given fraction r(e) of the matchings. Although it can be determined in polynomial time whether such a sequence of matchings exists or not [Grötschel et al., Combinatorica (1981), 169–197], we show that several questions about the length of the sequence are computationally intractable. Therefore, as is commonly done [Golumbic, Algorithmic graph theory and perfect graphs, 2004], we restrict our investigation to a special class of graphs. In recent work [Birand et al., INFOCOM 2010 Proceedings, 2010], two of the authors dealt with so-called OLoP (Overall Local Pooling) graphs, a class of graphs for which similar matching-related problems are tractable (namely, in an online distributed wireless network scheduling setting). We therefore focus on these graphs and generalize the results to a larger class of graphs which we call GOLoP graphs. In particular, we show that deciding whether a given GOLoP graph has a matching sequence of length at most k can be done in linear time. In case the answer is affirmative, we show how to construct, in quadratic time, the matching sequence of length at most k. Finally, we prove that, for GOLoP graphs, the length of a shortest sequence does not exceed a constant times the least common denominator of the fractions r(e), leading to a pseudopolynomial-time algorithm for minimizing the length of the sequence. We show that the constant equals 1 for OLoP graphs and, following Seymour [Seymour, Proc. London Math. Soc., 1979], conjecture that the constant is as small as 2 for general graphs. We then show that this conjecture holds for all graphs with at most 10 vertices. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol., 2013

We consider the maximum flow problem with minimum quantities (MFPMQ), which is a variant of the maximum flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound (a minimum quantity), which may depend on the arc. This problem has recently been shown to be weakly NP -complete even on series–parallel graphs. In this article, we provide further complexity and approximability results for MFPMQ and several special cases. We first show that it is strongly NP -hard to approximate MFPMQ on general graphs (and even bipartite graphs) within any positive factor. On series–parallel graphs, however, we present a pseudo-polynomial time dynamic programming algorithm for the problem. We then study the case that the minimum quantity is the same for each arc in the network and show that, under this restriction, the problem is still weakly NP -complete on general graphs, but can be solved in strongly polynomial time on series–parallel graphs. On general graphs, we present a -approximation algorithm for this case, where λ denotes the common minimum quantity of all arcs. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

Increased fuel costs together with environmental concerns have led shipping companies to consider the optimization of vessel speeds. Given a fixed sequence of port calls, each with a time window, and fuel cost as a convex function of vessel speed, we show that optimal speeds can be found in quadratic time. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

The paper “Multistage Methods for Freight Train Classification” by Jacob et al. [Networks 57 (2011) 87–105] provides great insight into the theory and practice of sorting procedures at shunting yards. In Jacob et al. [Networks 57 (2011) 87–105] many relevant shunting situations (e.g., single or multiple inbound trains, single or multiple outbound trains, (un)restricted number of tracks, (un)restricted track capacity) are formally specified as optimization problems. Then, for almost all of them either an exact polynomial-time algorithm or an NP-hardness proof is provided. However, the case of multiple inbound trains, which is of high practical relevance, is left open. We close this gap by providing a proof of NP-hardness. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

We present an extension to nonatomic congestion games (NCG). An NCG models a large number of players depending on a set of resources (e.g., network links) in certain combinations (e.g., paths or multicast trees) called strategies. The rate of consumption Z_eS specifies how aggressively resource e is consumed when used via strategy S, but it also effects how strongly the resource's latency is experienced by the players. Our extension allows essentially unrelated factors C_eS and R_eS instead of Z_eS. Factor C_eS is the actual rate of consumption, whereas R_eS expresses the amplification of the resource latency of e for players choosing strategy S, or, in other words, the relevance of resource e for strategy S. We call the extended model nonatomic consumption-relevance congestion games (NCRCG). NCRCGs exhibit new phenomena, including multiple Nash equilibria of different social cost and—even from a worst-case point of view—a dependence of the price of anarchy on structural parameters not limited to the class of resource latency functions used. We prove almost tight lower, upper, and bicriteria bounds for the price of anarchy for polynomial latency functions with nonnegative coefficients. We conjecture that the lower bound is the best possible. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

We consider the following simple network design problem. The input consists of n weighted nodes, and the output is an edge-weighted connected network such that the total weight of the edges incident to a node is at least the given weight of the node. We aim to design the cheapest connected network; that is, the reachability of the network should be guaranteed, and the network is better if its total weight is less. In this article, we first show an efficient algorithm that produces an optimal network with minimum weight. The algorithm runs in linear time, and the resulting network contains at most n edges, where n is the number of nodes. To construct a connected network, at least n - 1 edges are required. However, the algorithm sometimes outputs n edges. Next, we aim to minimize not only the weight but also the number of edges. That is, for given n weighted nodes, we aim to design a cheapest tree. Then, the problem becomes -complete. We also propose efficient approximation algorithms for constructing a cheapest tree. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

A multilayer network is a hierarchical network where each layer is built using the components of the previous one. Optical networks are an example of two layered networks. The multilayer network design problem consists of installing minimum cost integer capacities on the edges of all the layers so that a set of demands can be routed on the network. In this article, two versions of the optical network design problem are studied, and polyhedral results for the corresponding capacity formulations are presented. We also show how to extend the results to a network with an arbitrary number of layers. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Previous studies of search in channel graphs have assumed global search, for which the status of any link can be probed by the search algorithm at any time. We consider for the first time local search, for which only links to which an idle path from the source has already been established may be probed. We show that some well-known channel graphs may require exponentially more probes, on average, when search must be local than when it may be global. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

We study a variant of the shortest path problem in graphs: given a weighted graph Gand vertices sand t, and given a set Xof forbidden paths in G, find a shortest s- tpath Psuch that no path in Xis a subpath of P. Path Pis allowed to repeat vertices and edges. We call each path in Xan exception, and our desired path a shortest exception avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x∈Xonly when a path containing xfails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to use a shortest path algorithm incrementally after replicating vertices when an exception is discovered. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Motivated by the problem of link scheduling in wireless sensor networks where different sensors have different transmission and interference ranges and may be mobile, we study the problem of “distance edge coloring” of graphs, which is a generalization of proper edge coloring. Let Gbe a graph modeling a sensor network. An ℓ-distance edge coloring of Gis a coloring of the edges of Gsuch that any two edges within distance ℓof each other are assigned different colors. The parameter ℓis chosen, so that the links corresponding to two edges that are assigned the same color do not interfere. We investigate the ℓ-distance edge coloring problem on several families of graphs that can be used as topologies in sensor deployment. We focus on determining the minimum number of colors needed and optimal coloring algorithms. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Relocation of service-providers in response to changing real-time needs is suboptimal due to limited foreknowledge of client requests. Simple cost schedules for relocation and remote-service provision have been investigated both for the possibility of complete optimizability and the degree of inefficiency introduced by imperfect future knowledge. This work further explores a parametrization developed for reflecting limitations in the mobility of some resources. The optimizability response to this parameter exhibits two significant thresholds. Below the first threshold, optimization is trivial, but many real-world resource-location problems correspond to parameter values past the second threshold. This work explores both the value of the second threshold and the behavior of optimal limited-lookahead responses for resources whose immobility places them past this threshold. It is determined that, for resources of sufficiently high immobility α, it is possible with finite lookahead to achieve a relocation schedule which is within a ratio of (1+α) of the optimal omniscient schedule. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

In many applications of multiple objective network programming (MONP) problems, only integer solutions are acceptable as the final optimal solution. Representative efficient solutions are usually obtained by sampling the efficient set through the solution of augmented weighted Tchebycheff network programs. Because such efficient solutions are usually not integer solutions, a branch-and-bound (BB) algorithm is developed to find integer efficient solutions. The purpose of the BB algorithm is to support interactive procedures by generating representative integer efficient solutions. To be computationally efficient, the algorithm takes advantage of the network structure as much as possible. An algorithm, used in the BB algorithm and performed on the key tree, is developed to construct feasible solutions from infeasible solutions and basic solutions from nonbasic solutions when bounds on branching variables change. The BB algorithm finds basic and nonbasic or supported and unsupported integer efficient solutions as long as they are optimal. Details of the algorithm are presented, an example is provided and computational results are reported. Computational results show that the BB algorithm performs well. Although the BB algorithm is developed for the purpose of generating integer efficient solutions for MONP problems, it can also solve more general integer network flow problems with linear side constraints. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

This article focuses on the problem of computing a minimum-weight subgraph with unicyclic connected components. Although this problem is generally easy, it becomes difficult when a girth constraint is added. A polyhedral study is proposed. Many facets and valid inequalities are derived. Some of them can be exactly separated in polynomial time. Hence, the problem is solved by a cutting-plane algorithm based on these inequalities and using a compact formulation derived from the transversality of the bicircular matroid. Numerical results are also presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

We consider a network design problem in which flow bifurcations are allowed. The demand data are assumed to be uncertain, and the uncertainties of demands are expressed by an uncertainty set. The goal is to install facilities on the edges at minimum cost. The solution should be able to deliver any of the demand requirements defined in the uncertainty set. We propose an exact solution algorithm based on a decomposition approach in which the problem is decomposed into two distinct problems: (1) designing edge capacities; and (2) checking the feasibility of the designed edge capacities with respect to the uncertain demand requirements. The algorithm is a special case of the Benders decomposition method. We show that the robust version of the Benders subproblem can be formulated as a linear program whose size is polynomially bounded. We also propose a simultaneous cut generation scheme to accelerate convergence of the Benders decomposition algorithm. Computational results on real-life telecommunication problems are reported, and these demonstrate that robust solutions with very small penalties in the objective values can be obtained. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

This article analyzes inequalities derived by projecting out the flow variables of a single-commodity flow model for a vehicle routing problem. These inequalities are called reverse multistar (RMS) inequalities and are related to the MS inequalities analyzed and used in other articles. Although the MS RMS inequalities are irrelevant for some vehicle routing problems, in others they are fundamental. The article presents a vehicle routing problem in which the RMS are of interest. It is called the vehicle routing problem with lower and upper bound capacities (LU-VRP). It concerns a vehicle routing problem with one depot and a homogeneous fleet of vehicles. All the customers have a unit demand, and there are upper and lower bounds on the demand covered by each vehicle. New families of inequalities are derived by strengthening the RMS inequalities. Computational experiments show that the new inequalities are useful when solving LU-VRP instances. The experiments are based on variations of symmetric and asymmetric VRP library (VRPLIB) instances with up to 100 customers. The constraint on a minimum number of customers served in each route is implicit in the unit-demand capacitated vehicle routing problem with a fixed number of vehicles. Therefore, the article also evaluates the impact of using the lower bound inequalities developed in the context of this variant. It is still unknown whether the new inequalities can help solve it or not. Our theoretical analysis suggests that one of the families of developed inequalities is not implied by other standard inequalities. This prompts us to pursue studies in the search for other families of lower bound based inequalities for this variant. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

Theoretical analyses of a set of iterated-tour partitioning vehicle routing algorithms applicable to a wide variety of commonly used vehicle routing problem variants are presented. We analyze the worst-case performance of the algorithms and establish tightness of the derived bounds. Among other variants, we capture the cases of pick-up and delivery and multiple depots. We also introduce brand new concepts such as mobile depots, partitioning of customer nodes into groups, and potential opportunistic under-utilization of vehicle capacity by only partially loading the vehicle, among others, which arise from a printed circuit board application. The problems studied are of critical importance in many practical applications. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

The star graph proposed by Akers et al. (Proc Int Conf Parallel Process, University Park, PA, 1987, pp. 393–400) has many advantages over the n-cube. However, it suffers from having large gaps in the possible number of vertices. The arrangement graph was proposed by Day and Tripathi (Inf Process Lett 42 (1992), 235–241) to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. In fact, it also generalizes the alternating group graph (Jwo et al., Networks 23 (1993), 315–326). There are many different measures of structural integrity of interconnection networks. In this article, we prove results of the following type for the arrangement graph: If h(r,n,k) vertices are deleted from the arrangement graph A_n,k, the resulting graph will either be connected or have a large component and small components having at most r − 1 vertices in total. Our result is tight for r ≤ 3, and it is asymptotically tight for r ≥ 4. Moreover, we also determine the cyclic vertex-connectivity of the arrangement graph. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

We consider a rooted tree graph with costs associated with the edges and profits associated with the vertices. Every subtree containing the root incurs the sum of the costs of its edges, and collects the sum of the profits of its nodes; the goal is the simultaneous minimization of the total cost and maximization of the total profit. This problem is related to the TSP with profits on graphs with a tree metric. We analyze the problem from a biobjective point of view. We show that finding all extreme supported efficient points can be done in polynomial time. The problem of finding all efficient points, however, is harder; we propose a practical FPTAS for solving this problem. Some special cases are considered where the particular profit/cost structure or graph topology allows the efficient points to be found in polynomial time. Our results can be extended to more general graphs with distance matrices satisfying the Kalmanson conditions. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this article, we consider as sensors the set where one sensor can check its neighbors within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighborhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

Energy conservation is drawing increasing attention in data networking. As networks are designed for peak traffic, network elements typically operate at full speed and consume maximum power even when carrying low traffic. One school of thought believes that a dominant amount of power saving comes from turning off network elements. The difficulty is that transitioning between the active and sleeping modes consumes considerable energy and time. This results in an obvious trade-off between saving energy and provisioning performance guarantees such as end-to-end delays. We study the following routing and scheduling problem in a network in which each network element either operates in the full-rate active mode or the zero-rate sleeping mode. For a given network and traffic matrix, routing determines the path that each traffic stream traverses. For frame-based periodic scheduling, a schedule determines the active period per element within each frame and prioritizes packets within each active period. For a line topology, we present a schedule with close-to-minimum delay for a minimum active period per element. For an arbitrary topology, we partition the network into a collection of lines and use the near-optimal schedule along each line. Additional delay is incurred only when a path switches from one line to another. By minimizing the number of switchings via routing, we show a logarithmic approximation for both power consumption and end-to-end delays. If routing is given as input, we present two schedules one of which has active period proportional to the traffic load per network element, and the other has active period proportional to the maximum load over all elements. The end-to-end delay of the latter is much improved compared to the delay for the former. This demonstrates the trade-off between power and delay. Finally, we provide simulation results to validate our algorithmic approaches. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

We investigate the problem of designing a minimum-cost flow network interconnecting n sources and a single sink, each with known locations in a normed space and with associated flow demands. The network may contain any finite number of additional unprescribed nodes from the space; these are known as the Steiner points. For concave increasing cost functions, a minimum-cost network of this sort has a tree topology, and hence can be called a Minimum Gilbert Arborescence (MGA). We characterize the local topological structure of Steiner points in MGAs, showing, in particular, that for a wide range of metrics, and for some typical real-world cost functions, the degree of each Steiner point is 3. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

We consider an integrated job scheduling and network routing problem which appears in Grid Computing and production planning. The problem is to schedule a number of jobs at a finite set of machines, such that the overall profit of the executed jobs is maximized. Each job demands a number of resources which must be sent to the executing machine through a network with limited capacity. A job cannot start before all of its resources have arrived at the machine.

The scheduling problem is formulated as a Mixed Integer Program (MIP) and proved to be -hard. An exact solution approach using Dantzig-Wolfe decomposition is presented. The pricing problem is the linear multicommodity flow problem defined on a time-space network. Branching strategies are presented for the branch-and-price algorithm and three heuristics and an exact solution method are implemented for finding a feasible start solution. Finally, interior point stabilization is used to decrease the number of columns generated in the branch-and-price algorithm.

The algorithm is experimentally evaluated on job scheduling instances for a Grid network. The Dantzig-Wolfe algorithm with stabilization is clearly superior, being able to solve large instances with 1,000 jobs and 1,000 machines covering 24 hours of scheduling activity on a Grid network. The algorithm is also compared to simulations of a real-life Grid, and results show that the solution quality significantly increases when solving the problem to optimality. The promising results indicate that the algorithm can be used as an actual scheduling algorithm in the Grid or as a tool for analyzing Grid performance when adding extra machines or jobs. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

While most of today's research effort is being devoted to wireless technologies involving the tiniest and most sophisticated devices, Ethernet is evolving from a best effort, plug-and-play LAN technology, towards a carrier-grade WAN technology. Most of the new Ethernet standards rely on spanning tree protocols (STP) such as Rapid STP (RSTP) and multiple STP (MSTP). RSTP offers faster convergence than the legacy STP but like its predecessor, it uses a single tree to carry all the traffic offered to the network, seriously impacting throughput and bandwidth usage. MSTP however supports multiple spanning tree instances but does not provide generic methods to build those instances. Moreover, MSTP does not provide efficient methods to map between spanning trees and virtual LANs (VLAN). Operators must manually provision this mapping which seriously affects network operation expenditures and network performance. In this paper, we introduce a multiple spanning tree generation algorithm (MSTGA) and a VLAN-spanning tree mapping algorithm (VSTMA) aimed at helping operators leverage their networks, save bandwidth, and support service level agreements with their customers. These algorithms can be used to extend and/or work with MSTP. We show that MSTGA maximizes throughput while VSTMA minimizes bandwidth usage. We also show that combining edge-disjoint spanning trees with VSTMA constitutes the best bandwidth/throughput tradeoff. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013