Improved Approximation Algorithms for Capacitated Fault-Tolerant k-Center

TL;DR

This paper introduces improved approximation algorithms for the capacitated fault-tolerant k-center problem, achieving better approximation factors by using a novel backup center selection approach, and extends results to general capacities and related problems.

Contribution

The paper presents new approximation algorithms for the capacitated fault-tolerant k-center problem, improving previous bounds and addressing general capacity cases with a novel backup center strategy.

Findings

Achieved a 6-approximation for the basic capacitated fault-tolerant k-center problem.

Developed a 7-approximation for the conservative variant with limited reassignment.

Extended the approach to general capacities and related k-supplier problems.

Abstract

In the k-center problem, given a metric space V and a positive integer k, one wants to select k elements (centers) of V and an assignment from V to centers, minimizing the maximum distance between an element of V and its assigned center. One of the most general variants is the capacitated {\alpha}-fault-tolerant k-center, where centers have a limit on the number of assigned elements, and, if {\alpha} centers fail, there is a reassignment from V to non-faulty centers. In this paper, we present a new approach to tackle fault tolerance, by selecting and pre-opening a set of backup centers, then solving the obtained residual instance. For the {0,L}-capacitated case, we give approximations with factor 6 for the basic problem, and 7 for the so called conservative variant, when only clients whose centers failed may be reassigned. Our algorithms improve on the previously best known factors of 9 and 17, respectively. Moreover, we consider the case with general capacities. Assuming {\alpha} is constant, our method leads to the first approximations for this case. We also derive approximations for the capacitated fault- tolerant k-supplier problem.