We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times—the total latency—is minimized. In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a “selfishly motivated ” assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total

We describe four versions of a Greedy Randomized Adaptive Search Procedure (GRASP) for finding approximate solutions of general instances of the Steiner Problem in Graphs. Di#erent construction and local search algorithms are presented. Preliminary computational results with one of the versions on a variety of test problems are reported. On the majority of instances from the OR-Library, a set of standard test problems, the GRASP produced optimal solutions. On those that optimal solutions were not found, the GRASP found good quality approximate solutions.

We consider a class of network design problems in which one needs to find a minimum-cost network satisfying certain connectivity requirements. For example, in the survivable network design problem, the requirements specify that there should be at least r(v; w) edge-disjoint paths between each pair of vertices v and w. We present an approximation algorithm with a performance guarantee of 2H(fmax ) = 2(1 + 2 + 3 + \Delta \Delta \Delta + fmax ) where fmax is the maximum requirement. This improves upon the best previously known performance guarantee of 2fmax . We also show

Abstract We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency. A natural measure for the performance of a network used by selfish agents is the common latency experienced by each user in a Nash equilibrium. It is a counterintuitive but well-known fact that removing edges from a network may improve its performance; the most famous example of this phenomenon is the so-called Braess's Paradox. This fact motivates the following network design problem: given such a network, which edges should be removed to obtain the best possible flow at Nash equilibrium? Equivalently, given a large network of candidate edges to be built, which subnetwork will exhibit the best performance when used selfishly? We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n vertices and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = N P). We also prove this hardness result to be best possible by exhibiting an n/2-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no ( 43- ffl)-approximation algorithm for the problem (for any ffl> 0, unless P = N P); the existence of a 43-approximation algorithm follows easily from existing work, proving this hardness result sharp. Moreover, we prove that an optimal approximation algorithm for these problems is what we call the trivial algorithm: given a network of candidate edges, build the entire network. A consequence of this result is that Braess's Paradox (even in its worst-possible manifestation) is impossible to detect efficiently.

To efficiently derive bounds for large-scale instances of the capacitated fixed-charge network design problem, Lagrangian relaxations appear promising. This paper presents the results of comprehensive experiments aimed at calibrating and comparing bundle and subgradient methods applied to the optimization of Lagrangian duals arising from two Lagrangian relaxations. This study substantiates the fact that bundle methods appear superior to subgradient approaches because they converge faster and are more robust relative to different relaxations, problem characteristics, and selection of the initial parameter values. It also demonstrates that effective lower bounds may be computed efficiently for large-scale instances of the capacitated fixed-charge network design problem. Indeed, in a fraction of the time required by a standard simplex approach to solve the linear programming relaxation, the methods we present attain very high quality solutions.

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A decision support system for the management of an intermodal container terminal is presented. Among the problems to be solved, there are the spatial allocation of containers on the terminal yard, the allocation of resources and the scheduling of operations in order to maximise a performance function based on some economic indicators. These problems are solved using techniques from optimisation, like job-shop scheduling, genetic algorithms or mixed-integer linear programming. At the terminal, the same problems are usually solved by the terminal manager, only using his/her experience. The manager can trust computer generated solutions only by validating them by means of a simulation model of the terminal. Thus, the simulation tool also becomes a means to introduce new approaches into traditional settings. In the present paper we focus on the resource allocation problem. We describe our modules for the optimisation of the allocation process and for the simulation of the terminal. The former is based on integer linear programming; the latter is a discrete event simulation tool, based on the process-oriented paradigm. The simulator provides a test bed for checking the validity and the robustness of the policy computed by the optimisation module. The case study of the Contship La Spezia Container Terminal, located in the Mediterranean Sea in Italy, is examined. 1.

Real-world networks often need to be designed under uncertainty, with only partial information and predictions of demand available at the outset of the design process. The field of stochastic optimization deals with such problems where the forecasts are specified in terms of probability distributions of future data. In this paper, we broaden the set of models as well as the techniques being considered for approximating stochastic optimization problems. For example, we look at stochastic models where the cost of the elements is correlated to the set of realized demands, and risk-averse models where upper bounds are placed on the amount spent in each of the stages. These generalized models require new techniques, and our solutions are based on a novel combination of the primal-dual method truncated based on optimal LP relaxation values, followed by a treerounding stage. We use these to give constant-factor approximation algorithms for the stochastic Steiner tree and single sink network design problems in these generalized models. 1.

The bounded diameter minimum spanning tree (BDMST) problem is NP-hard and appears e.g. in communication network design when considering certain quality of service requirements. Prior exact approaches for the BDMST problem mostly rely on network flow-based mixed integer linear programming and Miller-Tucker-Zemlin-based formulations. This article presents a new, compact 0–1 integer linear programming model, which is further strengthened by dynamically adding violated connection and cycle elimination constraints within a branch-and-cut environment. The proposed approach is empirically compared to two recently published formulations. It turns out to work well in particular on dense instances with tight diameter bounds.

Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edge-cost-ow problems. An edge-cost ow problem is a min-cost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow.