Approximation Algorithm for Finding Protected Links to Keep Small Diameter against Link Failures

High reliability and performance are needed as the Internet becomes an important social infrastructure. Network delay is one of measures of performance of network. As delay between two nodes correlates roughly with the distance between them, the diameter of the network which is the maximum distance of all two nodes must be small. In addition, it is necessary to keep small network delay against network failures. Therefore, critical links whose failures significantly degrade the performance must be protected by rapid recovery so that failures cannot be detected over the IP layer. The number of these protected links must be small to restrict investment cost of facilities and operational cost for Internet service providers. Consequently, it is important to find the smallest number of links to be protected so that the diameter of a network is not more than a given integer even if a limited number of non-protected links fail. It was proved by the authors that this problem is NP-hard when the number of the simultaneous link failures, k, is more than or equal to three, and it can be solved in polynomial time when k is restricted to one. However, it was not known whether the problem is NP-hard or not when k = 2. Furthermore, no approximation algorithm was known for the NP-hard cases. In this paper, we solve these unsolved issues of this problem. First, we prove that the problem is NP-hard even if k = 2. Second, we present approximation algorithms with the approximation ratio of a fixed constant for the NP-hard cases. These results make clear the computational complexity of this problem theoretically and give the useful network design algorithms also from the practical viewpoints.