Detecting protected links to keep reachability to server against failures

Methods to design a reliable network against network failures are important for Internet service providers (ISPs). ISPs must offer connectivity to users, even if link failures occur. The best way to design a reliable network is to protect all links so that the failure probability of a link is sufficiently small by sufficient backup resource and rapid recovery system. However, sufficient backup resource needs much cost for ISP. Therefore, it is necessary to find the smallest number of links to be protected such that, even if any non-protected links fail, the resulting network offers users connectivity. In this paper, as a measure of connectivity, we focus on the size of the connected component including a server or a gateway. Critical links whose failures drastically decrease the size of the connected component including a server must be protected not to fail, because such a failure prevent a large number of users from access to the server. Similarly, when a server is a gateway to another ISP´s network, many users cannot communicate with exterior users and servers. Our purpose is to find the links to be protected such that, even if any non-protected links fail, the number of the nodes connected with a server in the resulting network is not less than a threshold. In this paper, we formulate this problem and prove that the problem is NP-hard. In addition, we propose a polynomial-time algorithm to solve the problem that the number of the simultaneous link failures is restricted to one. Furthermore, we present a polynomial-time algorithm to solve the problem that the number of the simultaneous link failures is restricted to two and a network is a tree.