Online Facility Location with Linear Delay

TL;DR

This paper introduces a new two-sided delay variant of the online facility location problem, providing an $O(1)$-competitive deterministic algorithm and extending techniques to the one-sided delay case with improved competitive ratios.

Contribution

It presents the first constant-competitive algorithm for the two-sided delay facility location problem and extends the approach to improve ratios for the one-sided delay variant.

Findings

Developed an $O(1)$-competitive deterministic algorithm for two-sided delay.

Extended the approach to achieve $O(rac{ ext{log} n}{ ext{log} ext{log} n})$-competitive ratio for one-sided delay.

Provided technical analysis based on a greedy strategy extending prior offline algorithms.

Abstract

We study the problem of online facility location with delay. In this problem, a sequence of $n$ clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a penalty: each client incurs a waiting cost (the difference between its arrival and connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. This is a well-studied problem both of its own, and within the general class of network design problems with delays. Our main focus is on a new variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay. We present an $O(1)$-competitive deterministic algorithm for the two-sided delay variant. On the technical side, we study a greedy strategy, which grows budgets with increasing waiting delays and opens facilities for subsets of clients once sums of these budgets reach certain thresholds. Our technique is a substantial extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location. We then show how to transform our $O(1)$-competitive algorithm for the two-sided delay variant to $O(\log n / \log \log n)$-competitive deterministic algorithm for one-sided delays. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios.