The backup 2‐center and backup 2‐median problems on trees

Abstract

In this paper, we are concerned with the problem of deploying two servers in a tree network, where each server may fail with a given probability. Once a server fails, the other server will take full responsibility for the services. Here, we assume that the servers do not fail simultaneously. In the backup 2-center problem, we want to deploy two servers at the vertices such that the expected distance from a farthest vertex to the closest functioning server is minimum. In the backup 2-median problem, we want to deploy two servers at the vertices such that the expected sum of distances from all vertices to the set of functioning servers is minimum. We propose an O(n)-time algorithm for the backup 2-center problem and an O(n log n)-time algorithm for the backup 2-median problem, where n is the number of vertices in the given tree network.

The 2-radius and 2-radiian problems on trees

“…For those edges removed from D, we compute all corresponding upper centers by a top-down approach. This problem is also solved in [25]. For completeness, we show how the approach works in the following.…”

Embeddings into almost self-centered graphs of given radius

“…The introduction of these graphs was in particular motivated with network designs in which two (expensive) resources that need to be far away due to interference reasons must be installed. Numerous additional situations appear in which two specific locations are desired, for instance in trees [10,20], block graphs [1,4], and interval graphs [9]. While in the first two of these papers 2-centers are studied in trees, the results could be used also in the general case.…”

Minimizing the makespan in multiserver network restoration problems

“…These cases are important because networks with tree or tree‐like topology often appear in practice, for example in production systems, time‐constrained delivery networks, and rural transportation systems , and exploiting this topology often makes it possible to obtain advanced algorithmic results. That is why extensive literature focuses on analytical results for various location, routing, and routing‐scheduling problems on trees (see, e.g., ] and the references therein) and cyclic networks with some tree‐like structure such as cactuses (e.g., ).…”