In highly reliable communication network design, disjoint paths between pairs of nodes are often needed in the design phase. The problem of finding k paths which are as diverse as possible and have the lowest total cost is called a k-best paths problem. We propose an algorithm for finding the k-best paths connecting a pair of nodes in a graph G. Graph extension is used to transfer the k-best paths problem to a problem which deploits well-known maximum flow (MaxFlow) and minimum cost network flow (MCNF) algorithms. We prove the k-best paths solution of our algorithm to be an optimal one and the time complexity is the same as MCNF algorithm. Our computational experiences show that the proposed algorithm can solve k-best paths problem for a large network within reasonable computation time.

In this paper, we consider the working VP and backup VP routing problems jointly and employ the integer programming based approach to maximize the system resource utilization and the network survivability. The VP planning problem is formulated as a nonlinear combinatorial optimization problem. The objective function minimizes the resource usage while maximizing the network survivability. By proper transformation of the objective function and applying cutting plane method, the original formulation is transformed into an integer linear programing formulation which is suitable for applying Lagrangian relaxation techniques. After Lagrangian relaxation, the problem is further decomposed into several tractable subproblems. Unlike others' work, the candidate path set does not need to be prepared in advance and the best paths are generated while solving subproblems in our approach. Heuristic algorithms based on the solving procedure of the Lagrangian relaxation are developed. Closely examining the gap between the heuristic upper bounds and the Lagrangian lower bounds reveals that the proposed algorithm can efficiently provide a nearly optimal solution for the survivable VP layout design in ATM networks.