Toward Online Mobile Facility Location on General Metrics

TL;DR

This paper introduces the online mobile facility location problem, provides lower bounds, and proposes algorithms for uniform metrics, advancing understanding of online facility location with mobile facilities.

Contribution

It formulates the OMFL problem, offers the first solution for a generalized uniform metric variant, and explores a new movement-cost-focused variant inspired by the k-server problem.

Findings

Established a lower bound for OMFL on uniform metrics.

Developed a deterministic online algorithm for G-OMFL with tight analysis.

Introduced M-OMFL, a variant focusing solely on movement costs, with a lower bound.

Abstract

We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. [SODA' 07]). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of $k$ mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics. A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the $k$-server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.