[Paper Review] Online Facility Location with Deletions

Marek Cygan, Artur Czumaj+2 more|arXiv (Cornell University)|Jul 10, 2018

Optimization and Search Problems37 references4 citations

TL;DR

This paper presents the first competitive online algorithms for the Facility Location problem with both client arrivals and deletions, introducing an optimal $O(\log n_{\text{act}}/\log\log n_{\text{act}})$-competitive algorithm for the uncapacitated variant and an $O(\log m + \log c \cdot \log n)$-competitive algorithm for the capacitated case, using hierarchically well-separated trees (HSTs) to manage dynamic reconnections and capacity constraints.

ABSTRACT

In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment. We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client's location. However, when a client who has been also one of the open facilities is to be removed, then our model has to allow to reconnect all clients that have been connected to that removed facility. In this model, we present an optimal O(log n_act / log log n_act)-competitive algorithm, where n_act is the number of active clients at the end of the input sequence. Next, we turn our attention to the capacitated Facility Location problem. We first note that if no deletions are allowed, then one can achieve an optimal competitive ratio of O(log n/ log log n), where n is the length of the sequence. However, when deletions are allowed, the capacitated version of the problem is significantly more challenging than the uncapacitated one. We show that still, using a more sophisticated algorithmic approach, one can obtain an online O(log m + log c log n)-competitive algorithm for the capacitated Facility Location problem in the fully dynamic model, where m is number of points in the input metric and c is the capacity of any open facility.

Motivation & Objective

• To address the gap in online Facility Location literature by studying fully dynamic scenarios where clients and facilities can both arrive and depart.
• To design efficient online algorithms that maintain low competitive ratios despite the complexity introduced by deletions.
• To extend existing randomized online algorithms for insertion-only settings to handle deletions while preserving near-optimal competitiveness.
• To analyze the capacitated variant under dynamic updates, where facility capacity constraints and client departures significantly increase algorithmic challenges.

Proposed method

• Adapt Meyerson's randomized insertion-only algorithm for uncapacitated Facility Location to handle deletions by allowing reassignment of clients from removed facilities.
• Use hierarchically well-separated trees (HSTs) to embed the original metric space with $O(\log m)$ expected stretch and $O(\log c)$ depth, enabling efficient distance estimation.
• Partition clients served by an optimal facility into levels based on their distance to the facility in the HST, and analyze cost contribution in two epochs: before and after a client becomes an open facility.
• Apply probabilistic analysis to bound the expected cost of reassignment during the second epoch, leveraging high-probability guarantees on facility availability within distance $O(\text{dist}_{\mathfrak{T}}(f, S_i))$.
• Decompose the total cost into components: facility opening cost, connection cost in HST, and reassignment penalties, each bounded via concentration and expectation arguments.
• Combine results from HST embedding and epoch-wise analysis to derive the final competitive ratio, showing that the dependence on $m$ and $c$ is logarithmic and optimal up to constants.

Experimental results

Research questions

• RQ1 Can an online algorithm achieve an optimal competitive ratio in the fully dynamic uncapacitated Facility Location problem where clients may arrive and depart?
• RQ2 What is the best possible competitive ratio for the capacitated Facility Location problem when deletions are allowed, and how does it compare to the insertion-only case?
• RQ3 Can the framework of hierarchically well-separated trees (HSTs) be effectively extended to handle dynamic client departures and facility reassignments in the capacitated setting?
• RQ4 Is it possible to maintain a competitive ratio close to the insertion-only bound in the presence of deletions, despite the increased complexity of reconnection and capacity constraints?
• RQ5 What is the trade-off between the number of points $m$, facility capacity $c$, and the competitive ratio in the dynamic capacitated model?

Key findings

• An $O(\log n_{\text{act}} / \log\log n_{\text{act}})$-competitive randomized algorithm is achieved for the fully dynamic uncapacitated Facility Location problem, matching the known optimal bound for the insertion-only case.
• For the capacitated variant with deletions, the paper presents an $O(\log m + \log c \cdot \log n)$-competitive algorithm, which is the first such result in this dynamic setting.
• The competitive ratio for the capacitated case is shown to be $O(\log m)$ in expectation for connection costs and $O(\log c \cdot \log n)$ for reassignment penalties, with the latter bounded via HST embedding and concentration bounds.
• The analysis shows that the expected cost of reassigning clients during the second epoch (after a client becomes a facility) is bounded by $O(\text{dist}_{\mathfrak{T}}(f, S_i)) + 120 \cdot \log c \cdot \ln n / c$, which contributes $O(\log c \cdot \log n)$ to the competitive ratio.
• The use of HSTs ensures $O(\log m)$ expected stretch and $O(\log c)$ depth, enabling efficient approximation of distances and facility availability in the dynamic setting.
• The paper demonstrates that despite the added complexity of deletions, both uncapacitated and capacitated variants admit online algorithms with competitive ratios matching or nearly matching the insertion-only bounds.