Robust and Survivable Network Design Considering Uncertain Node and Link Failures

Abstract

The network design is a planning process of placing system components to provide service or meet certain needs in an economical way. It has strong links to real application areas, such as transportation network, communication network, supply chain, power grid, water distribution systems, etc. In practice, these infrastructures are very vulnerable to any failures of system components. Therefore, the design of such infrastructure networks should be robust and survivable to any failures caused by many factors, for example, natural disasters, intentional attacks, system limits, etc. In this dissertation, we first summarize the background and motivations of our research topic on network design problems. Different from literature on network design, we consider both uncertain node and link failures during the network design process. The first part of our research is to design a survivable network with mixed connectivity requirements, or the (k,l)-connectivity. The designed network can still be connected after failures of any k vertices and (l-1) edges or failures of any (k-1) vertices and l edges. After formally proving its relationships to edge and vertex disjoint paths, we present two integer programming (IP) formulations, valid inequalities to strengthen the IP formulations, and a cutting plane algorithm. Numerical experiments are performed on randomly generated graphs to compare these approaches. Special cases of this problem include: when k=0, l=1, this problem becomes the well-known minimum spanning tree problem; and when k=0, l ≥ 1, this problem is to find a minimum-cost l-edge-connected spanning subgraph, while when k ≥ 2, l=0, the problem is to find a minimum-cost k-vertex-connected spanning subgraph. As a generalization of k-minimum spanning tree and λ-edge-connected spanning subgraph problems for network design, we consider the minimum-cost λ-edge-connected k-subgraph problem, or the (k, λ)-subgraph problem, which is to find a minimum-cost λ-edge-connected subgraph of a graph with at least k vertices. This problem can be considered as designing k-minimum spanning tree with higher connectivity requirements. We also propose several IP formulations for exactly solving the (k, λ)-subgraph problem, based on some graph properties, for example, requirements of cutsets for a division of the graph and paths between any two vertices. In addition, we study the properties of (k,2)-subgraphs, such as connectivity, bridgeless, and strong orientation properties. Based on these properties, we propose several stronger and more compact IP formulations for solving the (k,2)-subgraph problem, which is a direct generalization of the k-minimum spanning tree problem. Serving as a virtual backbone for wireless ad hoc networks, the connected dominating set problem has been widely studied. We design a robust and survivable connected dominating set for a virtual backbone of a larger graph for ad hoc network. More specifically, we study the (k,l)-connected d-dominating set problem. Given a graph G=(V,E), a subset D ⊆ V is a (k,l)-connected d-dominating set if the subgraph induced by D has mixed connectivity at least (k,l) and every vertex outside of S has at least d neighbors from D. The type of virtual backbone is survivable and also robust for sending message under certain number of both node and link failures. We study the properties of such dominating set and also IP formulations. In addition, we design a cutting plane algorithm to solve it.