# Reallocating Multiple Facilities on the Line

## TL;DR

This paper introduces a polynomial-time algorithm for the multistage K-facility reallocation problem on the real line, optimizing facility movements and agent connection costs over multiple stages, and presents an online algorithm for the case of two facilities.

## Contribution

It provides the first polynomial-time algorithm for optimal multistage K-facility reallocation and a constant-competitive online algorithm for two facilities, connecting to the K-server problem.

## Key findings

- Polynomial-time algorithm for optimal reallocation for any K.
- Constant-competitive online algorithm for K=2.
- Connection established with the classical K-server problem.

## Abstract

We study the multistage $K$-facility reallocation problem on the real line, where we maintain $K$ facility locations over $T$ stages, based on the stage-dependent locations of $n$ agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. $K$-facility reallocation was introduced by de Keijzer and Wojtczak, where they mostly focused on the special case of a single facility. Using an LP-based approach, we present a polynomial time algorithm that computes the optimal solution for any number of facilities. We also consider online $K$-facility reallocation, where the algorithm becomes aware of agent locations in a stage-by-stage fashion. By exploiting an interesting connection to the classical $K$-server problem, we present a constant-competitive algorithm for $K = 2$ facilities.