In this article, a heuristic is said to be *provably best* if, assuming
, no other heuristic always finds a better solution (when one exists). This extends the usual notion of “best possible” approximation algorithms to include a larger class of heuristics. We illustrate the idea on several problems that are somewhat stylized versions of real-life network optimization problems, including the maximum clique, maximum *k*-club, minimum (connected) dominating set, and minimum vertex coloring problems. The corresponding provably best construction heuristics resemble those commonly used within popular metaheuristics. Along the way, we show that it is hard to recognize whether the clique number and the *k*-club number of a graph are equal, yet a polynomial-time computable function is “sandwiched” between them. This is similar to the celebrated Lovász function wherein an efficiently computable function lies between two graph invariants that are
-hard to compute. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

The fast computation of point-to-point quickest paths on very large time-dependent road networks will allow next-generation web-based travel information services to take into account both congestion patterns and real-time traffic informations. The contribution of this article is threefold. First, we prove that, under special conditions, the Time-Dependent-Quickest Path Problem (QPP) can be solved as a static QPP with suitable-defined (constant) travel times. Second, we show that, if these special conditions do not hold, the static quickest path provides a heuristic solution for the original time-dependent problem with a worst-case guarantee. Third, we develop a time-dependent lower bound on the time-to-target which is both accurate and fast to compute. We show the potential of this bound by embedding it into a unidirectional algorithm which is tested on large metropolitan graphs. Computational results show that the new lower bound allows to reduce the computing time by 27% on average. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>This article introduces novel formulations for optimally responding to epidemics and cyber attacks in networks. In our models, at a given time period, network nodes (e.g., users or computing resources) are associated with probabilities of being infected, and each network edge is associated with some probability of propagating the infection. A decision maker would like to maximize the network's utility; keeping as many nodes open as possible, while satisfying given bounds on the probabilities of nodes being infected in the next time period. The model's relation to previous deterministic optimization models and to both probabilistic and deterministic asymptotic models is explored. Initially, maintaining the stochastic independence assumption of previous work, we formulate a nonlinear integer program with high-order multilinear terms. We then propose a quadratic formulation that provides a lower bound and feasible solution to the original problem. Further motivation for the quadratic model is given by showing that it alleviates the assumption of stochastic independence. The quadratic formulation is then linearized in order to be solved by standard integer programming solvers. We develop valid inequalities for the resulting formulations. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>We study variants of the vertex disjoint paths problem in plane graphs where paths have to be selected from given sets of paths. We investigate the problem as a decision, maximization, and routing-in-rounds problem. Although all considered variants are NP-hard in planar graphs, restrictions on the locations of the terminals on the outer face of the given planar embedding of the graph lead to polynomially solvable cases for the decision and maximization versions of the problem. For the routing-in-rounds problem, we obtain a *p*-approximation algorithm, where *p* is the maximum number of alternative paths for a terminal pair, when restricting the locations of the terminals to the outer face such that they appear in a counterclockwise traversal of the boundary as a sequence for some permutation . © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

Network routing games have attracted a lot of attention over the last years. However, most of the work assumes that the users have complete knowledge about the data in the network, in particular, about the precise travel times over the links. In this article, we address the more realistic case that users only know lower and upper bounds for the travel times and learn about the actual realization later. Thus, users cannot expect to choose a strategy which is optimal for each realization (scenario). This situation leads to combining concepts from robust optimization with algorithmic game theory. Specifically, we study the case where each user wants to minimize her regret, which is defined to be the worst-case difference of her bottleneck objective (the cost of her most expensive resource) to the optimum attainable value given *a priori* knowledge about the actual scenario. We show that in robust bottleneck routing games, equilibria do not always exist, but the existence can be guaranteed under the restriction that the problem is symmetric and the intervals of uncertainty have constant length. We prove that in general it is NP-hard to decide whether a given instance has a robust equilibrium. In the case of constant-length intervals of uncertainty, a robust equilibrium can be computed efficiently (in the symmetric case). We also investigate the Price of Robustness, which formally quantifies the lack of performance due to uncertainty and give a tight bound. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

With the increasing volume and volatility of Internet traffic, the need for adaptive routing algorithms has become compelling lately. An adaptive routing algorithm controls the rate at which traffic is placed on forwarding paths in concert with the actual user demands, making it possible to avoid congestion even when no information on expected traffic is available. In this article, we present a new model for rate-adaptive multipath routing, which allows one to analyze distributed, centralized, and hybrid routing architectures within a single framework, and to develop quantitative as well as qualitative arguments regarding their optimality, stability, and realizability. By a novel generalization of oblivious routing, we present a centralized algorithm with provable optimality, and we arrive at the conclusion that congestion can be completely eliminated even if routing decisions are completely precomputed. We find, although, that the complexity of the centralized scheme can become exponential. Therefore, we develop a hybrid distributed-centralized algorithm that combines the simplicity of distributed algorithms with the efficiency of centralized ones, and we provide numerical studies demonstrating that the hybrid scheme performs well in a broad selection of realistic scenarios. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>Vasko et al., Comput Oper Res 29 (2002), 441–458 defined the cable-trench problem (CTP) as a combination of the Shortest Path and Minimum Spanning Tree Problems. Specifically, let be a connected weighted graph with specified vertex (referred to as the *root*), length for each , and positive parameters and . The CTP is the problem of finding a spanning tree of such that is minimized, where is the total length of the spanning tree and is the total path length in from to all other vertices of . Recently, Jiang et al., Proceedings of MICCAI 6893 (2011), 528–536 modeled the vascular network connectivity problem in medical image analysis as an extraordinarily large-scale application of the generalized cable-trench problem (GCTP). They proposed an efficient solution based on a modification of Prim's algorithm (MOD_PRIM), but did not elaborate on it. In this article, we formally define the GCTP, describe MOD_PRIM in detail, and describe two linearly parallelizable metaheuristics which significantly improve the performance of MOD_PRIM. These metaheuristics are capable of finding near-optimal solutions of very large GCTPs in quadratic time in . We also give empirical results for graphs with up to 25,001 vertices.

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.

]]>The article studies the problem of designing telecommunication networks using transmission facilities of two different capacities. The point-to-point communication demands are met by installing a mix of facilities of both capacities on the edges to minimize total cost. We consider 3-partitions of the original graph which results in smaller 3-node subproblems. The extreme points of this subproblem polyhedron are characterized using a set of propositions. A new approach for computing the facets of the 3-node subproblem is introduced based on polarity theory. The facets of the subproblem are then translated back to those of the original problem using a generalized version of a previously known theorem. The approach has been tested on several randomly generated and real life networks. The computational results show that the new family of facets significantly strengthen the linear programming formulation and reduce the integrality gap. Also, there is a substantial reduction in the size of the branch-and-bound (B&B) tree if these facets are used. Problems as large as 37 nodes and 57 edges have been solved to optimality within a few minutes of computer time. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015

]]>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

]]>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

]]>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

]]>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, Vol. 65(4), 289–305 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, Vol. 65(4), 306–311 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, Vol. 65(4), 312–328 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. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 329–343 2015

In telecommunication networks, packets are carried from a source to a destination on a path determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from to follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore, the routing protocol must determine how the traffic from to is distributed among the shortest paths. In the protocol called Open Shortest Path First-Equal Cost Multiple Path (ospf-ecmp) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the “best” link lengths, toward a goal such as maximizing the network throughput for given link capacities. In this work, we show that the problem of maximizing *even a single* commodity flow for the ospf-ecmp protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 344–352 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 Vol. 65(4), 353–366 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, Vol. 65(4), 367–379 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, Vol. 65(4), 380–394 2015

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