A Constant Factor Approximation Algorithm for Fault-Tolerant k-Median

TL;DR

This paper introduces the first constant factor approximation algorithm for the fault-tolerant k-median problem, addressing both general and special cases, and extends results to fault-tolerant facility location with weighted service costs.

Contribution

It provides the first constant factor approximation for the general fault-tolerant k-median problem and polynomial algorithms for special graph classes, advancing the state of the art.

Findings

First constant factor approximation for general fault-tolerant k-median

Polynomial time algorithm for fault-tolerant k-median on paths and HSTs

Constant factor approximation for fault-tolerant facility location with weighted costs

Abstract

In this paper, we consider the fault-tolerant $k$-median problem and give the \emph{first} constant factor approximation algorithm for it. In the fault-tolerant generalization of classical $k$-median problem, each client $j$ needs to be assigned to at least $r_j \ge 1$ distinct open facilities. The service cost of $j$ is the sum of its distances to the $r_j$ facilities, and the $k$-median constraint restricts the number of open facilities to at most $k$. Previously, a constant factor was known only for the special case when all $r_j$s are the same, and a logarithmic approximation ratio for the general case. In addition, we present the first polynomial time algorithm for the fault-tolerant $k$-median problem on a path or a HST by showing that the corresponding LP always has an integral optimal solution. We also consider the fault-tolerant facility location problem, where the service cost of $j$ can be a weighted sum of its distance to the $r_j$ facilities. We give a simple constant factor approximation algorithm, generalizing several previous results which only work for nonincreasing weight vectors.