Motivated by a problem arising in the design of telecommunications networks using the SONET standard, we consider the problem of covering all edges of a graph using subgraphs that contain at most k edges with the objective of minimizing the total number of vertices in the subgraphs. We show that the problem is NP-hard when k 3 and present a linear-time -approximation algorithm. For even k values we present an approximation scheme with a reduced ratio but with increased complexity.

Abstract. An optimization problem arising in the design of optical fibre networks is discussed. A network contains client nodes, each installed on one or more SONET rings. A constraint programming model of the problem is described and compared with a mixed integer programming formulation. In the CP model the search is decomposed into two stages; first partially solving the problem by deciding how many rings each node should be on, and then making specific assignments of nodes to rings. The model includes implied constraints derived by considering optimal solutions to subproblems. In both the MIP and CP models, it is important to deal with the symmetry of the problem. In the CP model, two sources of symmetry are separated; one is eliminated dynamically during search and the other by assigning ranges rather than explicit values to one set of decision variables. The resulting CP model allows optimal solutions to be found easily for benchmark problems. 1

Mean streets represent those connected subsets of a spatial network whose attribute values are significantly higher than expected. Discovering and quantifying mean streets is an important problem with many applications such as detecting high-crime-density streets and high crash roads (or areas) for public safety, detecting urban cancer disease clusters for public health, detecting human activity patterns in asymmetric warfare scenarios, and detecting urban activity centers for consumer applications. However, discovering and quantifying mean streets in large spatial networks is computationally very expensive due to the difficulty of characterizing and enumerating the population of streets to define a norm or expected activity level. Previous work either focuses on statistical rigor at the cost of computational exorbitance, or

We develop a stochastic programming approach to solving an intra-ring Synchronous Optical Network (SONET) design problem. This research differs from pioneering SONET design studies in two fundamental ways. First, while traditional approaches to solving this problem assume that all data are deterministic, we observe that for practical planning situations, network demand levels are stochastic. Second, while most models disallow demand shortages and focus only on the minimization of capital Add-Drop Multiplexer (ADM) equipment expenditure, our model minimizes a mix of ADM installations and expected penalties arising from the failure to satisfy some or all of the actual telecommunication demand. We propose an L-shaped algorithm to solve this design problem, and demonstrate how a nonlinear reformulation of the problem may improve the strength of the generated optimality cuts. We next enhance the basic algorithm by implementing powerful lower and upper bounding techniques via an assortment of modeling, valid inequality, and heuristic strategies. Our computational results conclusively demonstrate the efficacy of our proposed algorithm as opposed to standard L-shaped and extensive form approaches to solving the problem. 1