Abstract not found

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-edge-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 worst-case analyses of two heuristics for the survivable network design problem.

We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst-case improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than ~. The corresponding factor for the class of clique tree inequalities is 8, while it is 4 for the path configuration inequalities. Keywords: Polyhedral combinatorics; Valid inequalities; Travelling salesman; Worst-case analysis 1.

Abstract not found

We consider the important practical and theoretical problem of designing a low-cost

This dissertation is the result of a project funded by Belgacom, the Belgian telecommunication operator, dealing with the development of new models and optimization techniques for the long-term planning of the backbone network. The minimum-cost two-connected spanning network problem consists in finding a network with minimal total cost for which there exist two node-disjoint paths between every pair of nodes. This problem, arising from the need to obtain survivable communication and transportation networks, has been widely studied. In our model, the following constraint is added in order to increase the reliability of the network : each edge must belong to a cycle of length less than or equal to a given threshold value K. This condition ensures that when traffic between two nodes has to be re-directed (e.g. in case of failure of an edge), we can limit the increase of the distance between these nodes. We investigate valid inequalities for this problem and provide numerical results obtai...

Designing low-cost networks that survive certain failure situations is one of the prime tasks in the telecommunication industry. In this paper we survey the development of models for network survivability used in practice in the last ten years. We show how algorithms integrating polyhedral combinatorics, linear programming, and various heuristic ideas can help solve real-world network dimensioning instances to optimality or within reasonable quality guarantees in acceptable running times. The most general problem type we address is the following. Let a communication demand between each pair of nodes of a telecommunication network be given. We consider the problem of choosing, among a discrete set of possible capacities, which capacity to install on each of the possible edges of the network in order to (i) satisfy all demands, (ii) minimize the building cost of the network. In addition to determining the network topology and the edge capacities we have to provide, for each demand, a routing such that (iii) no path can carry more than a given percentage of the demand, (iv) no path in the routing exceeds a given length. We also have to make sure that (v) for every single node or edge failure, a certain percentage of the demand is reroutable. Moreover, for all failure situations feasible routings must be computed. The model described above has been developed in cooperation with a German mobile phone provider. We present a mixed-integer programming formulation of this model and computational results with data from practice.

We study the approximability of dense and sparse instances of the following problems: the minimum 2-edge-connected (2-EC) and 2-vertex-connected (2-VC) spanning subgraph, metric TSP with distances 1 and 2 (TSP(1,2)), maximum path packing, and the longest path (cycle) problems. The approximability of dense instances of these problems was left open in Arora et al. [3]. We characterize the approximability of all these problems by proving tight upper (approximation algorithms) and lower bounds (inapproximability). We prove that 2-EC, 2-VC and TSP(1,2) are Max SNP-hard even on 3-regular graphs, and provide explicit hardness constants, under P 6= NP. We also improve the approximation ratio for 2-EC and 2-VC on graphs with maximum degree 3. These are the rst explicit hardness results on sparse and dense graphs for these problems. We apply our results to prove bounds on the integrality gaps of LP relaxations for dense and sparse 2-EC and TSP(1,2) problems, related to the famous metric TSP conjecture, due to Goemans [18].

We give a 17/12-approximation algorithm for the following NP-hard problem: Given an undirected graph, find a 2-edge connected spanning subgraph that has the minimum number of edges. The best previous approximation guarantee was 3/2. We conjecture that there is a 4/3-approximation algorithm. Thus our main result gets half-way to this target.

. Given a complete undirected graph with non-negative costs on the edges, the 2-Edge Connected Subgraph Problem consists in finding the minimum cost spanning 2-edge connected subgraph (where multiedges are allowed in the solution). A lower bound for the minimum cost 2-edge connected subgraph is obtained by solving the linear programming relaxation for this problem, which coincides with the subtour relaxation of the traveling salesman problem when the costs satisfy the triangle inequality. The simplest fractional solutions to the subtour relaxation are the 1 2 - integral solutions in which every edge variable has a value which is a multiple of 1 2 . We show that the minimum cost of a 2-edge connected subgraph is at most four-thirds the cost of the minimum cost 1 2 -integral solution of the subtour relaxation. This supports the long-standing 4 3 Conjecture for the TSP, which states that there is a Hamilton cycle which is within 4 3 times the cost of the optimal subtour relaxation ...