An optimal algorithm for the weighted backup 2-center problem on a tree

TL;DR

This paper introduces a linear time, optimal algorithm for solving the weighted backup 2-center problem on trees, which involves placing two facilities to minimize expected maximum distance considering failure probabilities and vertex weights.

Contribution

The paper presents the first linear time, asymptotically optimal algorithm for the weighted backup 2-center problem on trees using prune-and-search.

Findings

Algorithm runs in linear time

Achieves asymptotic optimality

Effectively handles weighted vertices and failure probabilities

Abstract

In this paper, we are concerned with the weighted backup 2-center problem on a tree. The backup 2-center problem is a kind of center facility location problem, in which one is asked to deploy two facilities, with a given probability to fail, in a network. Given that the two facilities do not fail simultaneously, the goal is to find two locations, possibly on edges, that minimize the expected value of the maximum distance over all vertices to their closest functioning facility. In the weighted setting, each vertex in the network is associated with a nonnegative weight, and the distance from vertex $u$ to $v$ is weighted by the weight of $u$. With the strategy of prune-and-search, we propose a linear time algorithm, which is asymptotically optimal, to solve the weighted backup 2-center problem on a tree.