Topology-Constrained Network Design

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Extended formulation for hop constrained distribution network configuration problems

International audienceA distribution network is a system aiming to transfer a certain type of resource from feeders to customers. Feeders are producers of a resource and customers have a certain demand in this resource that must be satisfied. Distribution networks can be represented on graphs and be subject to constraints that limit the number of intermediate nodes between some elements of the network (hop constraints) because of physical constraints. This paper uses layered graphs for hop constrained problems to build extended formulations. Preprocessing techniques are also presented to reduce the size of the layered graphs used. The presented model is studied on the hop-constrained minimum margin problem in an electricity network. This problem consists of designing a connected electricity distribution network, and to assign customers to electricity feeders at a maximum number of hops H so as to maximize the minimum capacity margin over the feeders to avoid an overload for any feeder. Numerical results of our model are compared with those of state-of-the-art solution techniques of the minimum margin problem form Rossi et al. [20]. Variations of the initial problem are also presented, considering losses due to transportation or by replacing hop constraints by distance constraints, a variation arising in the context of multicast transmission in telecommunications

Compact mixed integer linear programming models to the Minimum Weighted Tree Reconstruction problem

The Minimum Weighted Tree Reconstruction (MWTR) problem consists of finding a minimum length weighted tree connecting a set of terminal nodes in such a way that the length of the path between each pair of terminal nodes is greater than or equal to a given distance between the considered pair of terminal nodes. This problem has applications in several areas, namely, the inference of phylogenetic trees, the modeling of traffic networks and the analysis of internet infrastructures. In this paper, we investigate the MWTR problem and we present two compact mixed-integer linear programming models to solve the problem. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that the best of the two models is able to solve instances of the problem having up to 15 terminal nodes

Preface: Recent advances in telecommunications networks planning and operation

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Optimal design of switched Ethernet networks implementing the Multiple Spanning Tree Protocol

International audienceSwitched Ethernet networks rely on the Spanning Tree Protocol (STP) to ensure a cycle-free connectivity between nodes, by reducing the topology of the network to a spanning tree. The Multiple Spanning Tree Protocol (MSTP) allows for the providers to partition the traffic in the network and assign it to different virtual local area networks, each satisfying the STP. In this manner, it is possible to make a more efficient use of the physical resources in the network. In this paper we consider the traffic engineering problem of finding optimal designs of switched Ethernet networks implementing the MSTP, such that the worst-case link utilization is minimized. We show that this problem is N P-hard. We propose three mixed-integer linear programming formulations for this problem. Through a large set of computational experiments, we compare the performance of these formulations. Until now, the problem was almost exclusively solved with heuristics. Our objective here is provide a first comparison of different models that can be used in exact methods

Models for the piecewise linear unsplittable multicommodity flow problems

International audienceIn this paper, we consider multicommodity flow problems, with unsplit-table flows and piecewise linear routing costs. We first focus on the case where the piecewise linear routing costs are convex. We show that this problem is N P-hard for the general case, but polynomially solvable when there is only one commodity. We then propose a strengthened mixed-integer programming formulation for the problem. We show that the linear relaxation of this formulation always gives the optimal solution of the problem for the single commodity case. We present a wide array of computational experiments, showing this formulation also produces very tight linear programming bounds for the multi-commodity case. Finally, we also adapt our formulation for the non-convex case. Our experimental results imply that the linear programming bounds for this case, are only slightly than the ones of state-of-the-art models for the splittable flow version of the problem

Novel formulations for general and security Stackelberg games

In this paper we analyze general Stackelberg games (SGs) and Stackelberg security games (SSGs). SGs are hierarchical adversarial games where players select actions or strategies to optimize their payoffs in a sequential manner. SSGs are a type of SGs that arise in security applications, where the strategies of the player that acts first consist in protecting subsets of targets and the strategies of the followers consist in attacking one of the targets. We review existing mixed integer optimization formulations in both the general and the security setting and present new formulations for both settings. We compare the SG formulations and the SSG formulations both from a theoretical and a computational point of view. Our theoretical results show that the new formulations provide tighter linear relaxations. Our computational experiments show that the new formulations give better solution times

A study of general and security Stackelberg game formulations

International audienceIn this paper, we analyze different mathematical formulations for general Stackelberg games (GSGs) and Stackelberg security games (SSGs). We consider GSGs in which a single leader commits to a utility maximizing strategy knowing that one of p possible followers optimizes its own utility taking this leader strategy into account. SSGs are a type of GSG that arise in security applications where the strategies of the leader consist in protecting subsets of targets and the strategies of the p followers consist in attacking a single target. We compare existing mixed integer linear programming (MILP) formulations for GSGs, sorting them according to the tightness of their linear programming (LP) relaxations. We show that SSG formulations are projections of GSG formulations and exploit this link to derive a new SSG MILP formulation that i) has the tightest LP relaxation known among SSG MILP formulations and ii) its LP relaxation coincides with the convex hull of feasible solutions in the case of a single follower. We present computational experiments empirically comparing the difficulty of solving the formulations in the general and security settings. The new SSG MILP formulation is computationally efficient, in particular as the problem size increases

Benders decomposition for network design covering problems

Article number 105417We consider two covering variants of the network design problem. We are given a set of origin/destination pairs, called O/D pairs, and each such O/D pair is covered if there exists a path in the network from the origin to the destination whose length is not larger than a given threshold. In the first problem, called the Maximal Covering Network Design problem, one must determine a network that maximizes the total fulfilled demand of the covered O/D pairs subject to a budget constraint on the design costs of the network. In the second problem, called the Partial Covering Network Design problem, the design cost is minimized while a lower bound is set on the total demand covered. After presenting formulations, we develop a Benders decomposition approach to solve the problems. Further, we consider several stabilization methods to determine Benders cuts as well as the addition of cut-set inequalities to the master problem. We also consider the impact of adding an initial solution to our methods. Computational experiments show the efficiency of these different aspects.Feder (UE) PID2019- 106205GB-I00FEDER(UE) MTM2015-67706-PFonds de la Recherche Scientifique PDR T0098.1