Mobile Facility Location: Combinatorial Filtering via Weighted Occupancy Amitai Armon∗
Iftah Gamzu†
Danny Segev‡
Introduction Facility location problems have received a great deal of attention in the computer science and operations research communities, as they play an instrumental role in combinatorial optimization, supply chain management, and logistics. While most computational tasks in this increasingly-popular research field are directly motivated by practical scenarios, only a handful of these real-life models have also been established as useful tools in developing new algorithmic techniques or in understanding the limits of tractability. One of the most fundamental such models is the k-median problem, in which we would like to locate a certain number of facilities in a given network, trying to minimize the overall service cost of residing clients. In this paper, rather than considering the formation of such networks from scratch, we investigate the computational aspects of modifying an already existing network. An instance of the mobile facility location problem consists of a complete directed graph G = (V, E), in which each arc (u, v) ∈ E is associated with a numerical attribute M(u, v), representing the cost of moving any object from u to v. An additional ingredient of the input is a collection of k servers, S = {s1 , . . . , sk }, and a set of ` clients, C = {c1 , . . . , c` }, which are located at the nodes of the underlying graph. For simplicity of presentation, we overload existing notation, and interchangeably make use of si to denote the origin node of this server, noting that similar conventions will be utilized to denote the initial position of each client. With these definitions in mind, a movement scheme is a function ψ : S → V that relocates each server si to a new position, ψ(si ). In what follows, we refer to M(si , ψ(si )) as the relocation cost of this server, and to mini∈[k] M(cj , ψ(si )), the cost of assigning client cj to the nearest final server location, as the service cost of cj . The objective is to devise a movement scheme that minimizes the sum of relocation and service costs. The task of characterizing the approximability of mobile facility location was posed as an open question by Demaine et al. [2], who initiated the study of “movement problems”. In such settings, the goal is to relocate various objects into a final configuration, satisfying given design criteria, in a way that optimizes some movement-related measure. From this point of view, the mobile facility location problem involves two types of objects (servers and clients), desired configurations have to collocate every client with some server, and the objective is to minimize the total movement cost. Very recently, Friggstad and Salavatipour [3] studied the metric version of mobile facility location, and suggested a constant-factor approximation algorithm based on extending the k-median heuristic of Charikar, Guha, Tardos and Shmoys [1]. Nevertheless, approximating the mobile facility location problem in its unconfined form, where one makes no structural assumptions about the underlying graph and movement costs, has remained a challenging research objective. ∗
School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email:
[email protected]. School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email:
[email protected]. Supported by the Binational Science Foundation, and by the Israel Science Foundation. ‡ Department of Computer Science, Carnegie Mellon University, Pittsburgh PA 15213, USA. Email:
[email protected]. †
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Results and Technical Highlights In this paper, we resolve the above-mentioned open question by fully characterizing the approximability of mobile facility location through LP-based methods. At the same time, we make a concentrated effort to design a truly efficient and easy-to-implement combinatorial approach, which is of independent interest, as it may be applicable in other settings as well. Our findings, and the techniques by which we derive them, can be briefly summarized as follows. Tight approximability bounds. We devise a randomized LP-rounding algorithm that constructs, with constant probability, a movement scheme whose cost is within a factor of O(log `) of optimal. However, rather than constructing a feasible solution, this movement scheme deploys O(log `) copies of each server, meaning that every server is initially split into O(log `) identical duplicates, which may be independently relocated. Even though such resource augmentation may appear to be redundant at first glance, we proceed by demonstrating that this bi-criteria outcome is best possible, unless P = NP. Combinatorial bounds. As previously mentioned, motivated by real-life considerations, we propose a randomized combinatorial algorithm that constructs, with constant probability, a movement scheme whose cost is within a factor of O(log ` log k) of optimal. In this case, O(log ` log k) copies of each server are deployed. Combinatorial filtering via weighted occupancy. Technically speaking, the aforementioned algorithms are inspired by the filtering method, originally suggested by Lin and Vitter [5]. In the context of facility location problems, this technique utilizes fractional LP solutions to filter out costly service connections, by defining a nearby neighborhood for each client in which facilities will be located later on. Consequently, to come up with an efficient combinatorial approach, the main algorithmic challenge is to design a provably-good filtering method without solving any linear program. For this purpose, we introduce a weighted version of the occupancy problem [4], for which we establish surprising tail bounds, not before demonstrating that existing bounds cannot be extended. We believe that this newly-defined variant and its analysis may be interesting on their own right.
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