# Tight Approximability of the Server Allocation Problem for Real-Time Applications

## Abstract

The server allocation problem is a facility location problem for a distributed processing scheme on a real-time network. In this problem, we are given a set of users and a set of servers. Then, we consider the following communication process between users and servers. First a user sends his/her request to the nearest server. After receiving all the requests from users, the servers share the requests. A user will then receive the data processed from the nearest server. The goal of this problem is to choose a subset of servers so that the total delay of the above process is minimized. In this paper, we prove the following approximability and inapproximability results. We first show that the server allocation problem has no polynomial-time approximation algorithm unless **P** = **NP**. However, assuming that the delays satisfy the triangle inequality, we design a polynomial-time \({3 \over 2}\)-approximation algorithm. When we assume the triangle inequality only among servers, we propose a polynomial-time 2-approximation algorithm. Both of the algorithms are tight in the sense that we cannot obtain better polynomial-time approximation algorithms unless **P** = **NP**. Furthermore, we evaluate the practical performance of our algorithms through computational experiments, and show that our algorithms scale better and produce comparable solutions than the previously proposed method based on integer linear programming.

## Notes

### Acknowledgments

We thank Eiji Oki for bringing the problem into our attention.

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