Distance Transformation for Network Design Problems

International audienceWe propose a new generic way to construct extended formulations for a large class of network design problems with given connectivity requirements. The approach is based on a graph transformation that maps any graph into a layered graph according to a given distance function. The original connectivity requirements are in turn transformed into equivalent connectivity requirements in the layered graph. The mapping is extended to the graphs induced by fractional vectors through an extended linear integer programming formulation. While graphs induced by binary vectors are mapped to isomorphic layered graphs, those induced by fractional vectors are mapped to a set of graphs having worse connectivity properties. Hence, the connectivity requirements in the layered graph may cut off fractional vectors that were feasible for the problem formulated in the original graph. Experiments over instances of the Steiner Forest and Hop-constrained Survivable Network Design problems show that significant gap reductions over the state-of-the art formulations can be obtained

IP modeling of the survivable hop constrained connected facility location problem

Abstract We consider a generalized version of the rooted connected facility location problem which occurs in planning of telecommunication networks with both survivability and hop-length constraints. Given a set of client nodes, a set of potential facility nodes including one predetermined root facility, a set of optional Steiner nodes, and the set of the potential connections among these nodes, that task is to decide which facilities to open, how to assign the clients to the open facilities, and how to interconnect the open facilities in such a way, that the resulting network contains at least λ edge-disjoint paths, each containing at most H edges, between the root and each open facility and that the total cost for opening facilities and installing connections is minimal. We study two IP models for this problem and present a branch-and-cut algorithm based on Benders decomposition for finding its solution. Finally, we report computational results

Topology-Constrained Network Design

International audienc

A Flexible, Natural Formulation for the Network Design Problem with Vulnerability Constraints

Given a graph, a set of origin-destination (OD) pairs with communication requirements, and an integer k >= 2, the network design problem with vulnerability constraints (NDPVC) is to identify a subgraph with the minimum total edge costs such that, between each OD pair, there exist a hop-constrained primary path and a hop-constrained backup path after any k - 1 edges of the graph fail. Formulations exist for single-edge failures (i.e., k = 2). To solve the NDPVC for an arbitrary number of edge failures, we develop two natural formulations based on the notion of length-bounded cuts. We compare their strengths and flexibilities in solving the problem for k >= 3. We study different methods to separate infeasible solutions by computing length-bounded cuts of a given size. Experimental results show that, for single-edge failures, our formulation increases the number of solved benchmark instances from 61% (obtained within a two-hour limit by the best published algorithm) to more than 95%, thus increasing the number of solved instances by 1,065. Our formulation also accelerates the solution process for larger hop limits and efficiently solves the NDPVC for general k. We test our best algorithm for two to five simultaneous edge failures and investigate the impact of multiple failures on the network design

A branch-and-Benders-cut method for nonlinear power design in green wireless local area networks

We consider a problem arising in the design of green wireless local area networks. Decisions on powering-on a set of access points (APs), via the assignment of one power level (PL) to each opened AP, and decisions on the assignment of the user terminals (UTs) to the opened APs, have to be taken simultaneously. The PL assigned to an AP affects, in a nonlinear way, the capacity of the connections between the AP and the UTs that are assigned to it. The objective is to minimize the overall power consumption of the APs, which has two components: location/capacity dimensioning costs of the APs; assignment costs that depend on the total demands assigned to the APs. We develop a branch-and-Benders-cut (BBC) method where, in a non-standard fashion, the master problem includes the variables of the Benders subproblem, but relaxes their integrality. The BBC method has been tested on a large set of instances, and compared to a Benders decomposition algorithm on a subset of instances without assignment costs, where the two approaches can be compared. The computational results show the superiority of BBC in terms of solution quality, scalability and robustness

Robust capacitated trees and networks with uniform demands

We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in case of terminals with uniform demands. Formally, we are given a graph, capacity and cost functions on the edges, a root, a subset of nodes called terminals, and a bound k on the number of edge failures. We first study the problem where k = 1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Second, we consider the problem of computing a minimum-cost survivable network, i.e., a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set based formulation, a flow based one, and a bilevel one (with an attacker and a defender). We propose algorithms to solve each formulation and compare their efficiency

Single‐commodity stochastic network design under demand and topological uncertainties with insufficient data

Stochastic network design is fundamental to transportation and logistic problems in practice, yet faces new modeling and computational challenges resulted from heterogeneous sources of uncertainties and their unknown distributions given limited data. In this article, we design arcs in a network to optimize the cost of single‐commodity flows under random demand and arc disruptions. We minimize the network design cost plus cost associated with network performance under uncertainty evaluated by two schemes. The first scheme restricts demand and arc capacities in budgeted uncertainty sets and minimizes the worst‐case cost of supply generation and network flows for any possible realizations. The second scheme generates a finite set of samples from statistical information (e.g., moments) of data and minimizes the expected cost of supplies and flows, for which we bound the worst‐case cost using budgeted uncertainty sets. We develop cutting‐plane algorithms for solving the mixed‐integer nonlinear programming reformulations of the problem under the two schemes. We compare the computational efficacy of different approaches and analyze the results by testing diverse instances of random and real‐world networks. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 154–173, 2017Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/137236/1/nav21739_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/137236/2/nav21739.pd

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