Survivable Networks, Linear Programming Relaxations and the Parsimonious Property

We consider the survivable network design problem - the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem

Matroidal Degree-Bounded Minimum Spanning Trees

We consider the minimum spanning tree (MST) problem under the restriction that for every vertex v, the edges of the tree that are adjacent to v satisfy a given family of constraints. A famous example thereof is the classical degree-constrained MST problem, where for every vertex v, a simple upper bound on the degree is imposed. Iterative rounding/relaxation algorithms became the tool of choice for degree-bounded network design problems. A cornerstone for this development was the work of Singh and Lau, who showed for the degree-bounded MST problem how to find a spanning tree violating each degree bound by at most one unit and with cost at most the cost of an optimal solution that respects the degree bounds. However, current iterative rounding approaches face several limits when dealing with more general degree constraints. In particular, when several constraints are imposed on the edges adjacent to a vertex v, as for example when a partition of the edges adjacent to v is given and only a fixed number of elements can be chosen out of each set of the partition, current approaches might violate each of the constraints by a constant, instead of violating all constraints together by at most a constant number of edges. Furthermore, it is also not clear how previous iterative rounding approaches can be used for degree constraints where some edges are in a super-constant number of constraints. We extend iterative rounding/relaxation approaches both on a conceptual level as well as aspects involving their analysis to address these limitations. This leads to an efficient algorithm for the degree-constrained MST problem where for every vertex v, the edges adjacent to v have to be independent in a given matroid. The algorithm returns a spanning tree T of cost at most OPT, such that for every vertex v, it suffices to remove at most 8 edges from T to satisfy the matroidal degree constraint at v

The k-edge connected subgraph problem: Valid inequalities and Branch-and-Cut

International audienceIn this paper we consider the k-edge connected subgraph problem from a polyhedral point of view. We introduce further classes of valid inequalities for the associated polytope, and describe sufficient conditions for these inequalities to be facet defining. We also devise separation routines for these inequalities, and discuss some reduction operations that can be used in a preprocessing phase for the separation. Using these results, we develop a Branch-and-Cut algorithm and present some computational results

Strong Formulations for Network Design Problems with Connectivity Requirements

The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edge-connectivity NDC problem: unitary problems requiring connected network designs, and nonunitary problems permitting non-connected networks as solutions. We (i) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation, (ii) project out several classes of valid inequalities-partition inequalities, odd-hole inequalities, and combinatorial design inequalities-that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem, and (iii) show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the strength and power of flow-based formulations for network design problems with connectivity requirements

Optimization in Telecommunication Networks

Network design and network synthesis have been the classical optimization problems intelecommunication for a long time. In the recent past, there have been many technologicaldevelopments such as digitization of information, optical networks, internet, and wirelessnetworks. These developments have led to a series of new optimization problems. Thismanuscript gives an overview of the developments in solving both classical and moderntelecom optimization problems.We start with a short historical overview of the technological developments. Then,the classical (still actual) network design and synthesis problems are described with anemphasis on the latest developments on modelling and solving them. Classical results suchas Menger’s disjoint paths theorem, and Ford-Fulkerson’s max-flow-min-cut theorem, butalso Gomory-Hu trees and the Okamura-Seymour cut-condition, will be related to themodels described. Finally, we describe recent optimization problems such as routing andwavelength assignment, and grooming in optical networks.operations research and management science;

An Exact Algorithm for Robust Network Design

Modern life heavily relies on communication networks that operate efficiently. A crucial issue for the design of communication networks is robustness with respect to traffic fluctuations, since they often lead to congestion and traffic bottlenecks. In this paper, we address an NP-hard single commodity robust network design problem, where the traffic demands change over time. For k different times of the day, we are given for each node the amount of single-commodity flow it wants to send or to receive. The task is to determine the minimum-cost edge capacities such that the flow can be routed integrally through the net at all times. We present an exact branch-and-cut algorithm, based on a decomposition into biconnected network components, a clever primal heuristic for generating feasible solutions from the linear-programming relaxation, and a general cutting-plane separation routine that is based on projection and lifting. By presenting extensive experimental results on realistic instances from the literature, we show that a suitable combination of these algorithmic components can solve most of these instances to optimality. Furthermore, cutting-plane separation considerably improves the algorithmic performance