Fast approximation algorithms for the generalized survivable network design problem

In a standard f-connectivity network design problem, we are given an undirected graph G = (V, E), a cut-requirement function f : 2V → N, and non-negative costs c(e) for all e ∈ E. We are then asked to find a minimum-cost vector x ∈ ℕE such that x(δ(S)) ≥ f(S) for all S ⊆ V. We focus on the class of such problems where f is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the f-connectivity problem with general proper functions f. Implementing Jain's algorithm, this yields a strongly polynomial time (2 + ε)-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation)

Dynamic Network Topologies

Demand for effective network defense capabilities continues to increase as cyber attacks occur more and more frequently and gain more and more prominence in the media. Current security practices stop after data encryption and network address filtering. Security at the lowest level of network infrastructure allows for greater control of how the network traffic flows around the network. This research details two methods for extending security practices to the physical layer of a network by modifying the network infrastructure. The first method adapts the Advanced Encryption Standard while the second method uses a Steiner tree. After the network connections are updated, the traffic is re-routed using an approximation algorithm to solve the resulting multicommodity flow problem. The results show that modifying the network connections provides additional security to the information. Additionally, this research extends on previous research by addressing enterprise-size networks; networks between 5 and 1000 nodes with 1 through 5 interfaces are tested. While the final configuration depends greatly on the starting network infrastructure, the speed of the execution time enables administrators to make infrastructure adjustments in response to active cyber attacks

Survivable Logical Topology Mapping under Multiple Constraints in IP-over-WDM Networks

The survivable logical topology mapping problem in an IP-over-WDM network deals with the cascading effect of link failures from the bottom (physical) layer to the upper (logical) layer. Multiple logical links may get disconnected due to a single physical link failure, which may cause the disconnection of the logical network. Here we study survivability issues in IP-over-WDM networks with respect to various criteria.We first give an overview of the two major lines of pioneering works for the survivable design problem. Though theoretically elegant, the first approach which uses Integer Linear Programming (ILP) formulations suffers from the drawback of scalability. The second approach, the structural approach, utilizes the concept of duality between circuits and cutsets in a graph and is based on an algorithmic framework called Survivable Mapping Algorithm by Ring Trimming (SMART). Several SMART-based algorithms have been proposed in the literature.In order to generate the survivable routing, the SMART-based algorithms require the existence of disjoint lightpaths for certain groups of logical links in the physical topology, which might not always exist. Therefore, we propose in Chapter 4 an approach to augment the logical topology with new logical links to guarantee survivability. We first identify a logical topology that admits a survivable mapping against one physical link failure. We then generalize these results to achieve augmentation of a given logical topology to survive multiple physical link failures.We propose in Chapter 5 a generalized version of SMART-based algorithms and introduce the concept of robustness of an algorithm which captures the ability of the algorithm to provide survivability against multiple physical link failures. We demonstrate that even when a SMART-based algorithm cannot be guaranteed to provide survivability against multiple physical link failures, its robustness could be very high.Most previous works on the survivable logical topology design problem in IP-over-WDM networks did not consider physical capacities and logical demands. In Chapter 6, we study this problem taking into account logical link demands and physical link capacities. We define weak survivability and strong survivability in capacitated IP-over-WDM networks. Two-stage Mixed-Integer Linear Programming (MILP) formulations and heuristics to solve the survivable design problems are proposed. Based on the 2-stage MILP framework, we also propose several extensions to the weakly survivable design problem, considering several performance criteria. Noting that for some logical networks a survivable mapping may not exist, which prohibits us from applying the 2-stage MILP approach, our first extension is to augment the logical network using an MILP formulation to guarantee the existence of a survivable routing. We then propose approaches to balance the logical demands satisfying absolute or ratio-weighted fairness. Finally we show how to formulate the survivable logical topology design problem as an MILP for the multiple failure case.We conclude with an outline of two promising new directions of research

Algorithms and complexity analyses for some combinational optimization problems

The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design. In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios. In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs. Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost

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