Connected Facility Leasing Problems

Anais do Encontro de Teoria da Computação (ETC), 2018

In the leasing optimization model, resources are leased for K different time periods, instead of being acquired for unlimited duration. The goal is to use these temporary resources to maintain a dynamic infrastructure that serves n requests while minimizing the total cost. We propose and study a leasing variant of the online connected facility location problem, which we call the online connected facility leasing problem. In this problem each client that arrives must be connected to a temporary facility, which in turn must be connected to a root facility using permanent edges. We present an algorithm that is O(K · lg n)-competitive if the scaling factor is M = 1.

2007

We study the Connected Facility Location problems. We are given a connected graph G = (V, E) with non-negative edge cost c e for each edge e ∈ E, a set of clients D ⊆ V such that each client j ∈ D has positive demand d j and a set of facilities F ⊆ V each has non-negative opening cost f i and capacity to serve all client demands. The objective is to open a subset of facilities, say \(\hat F\) , to assign each client j ∈ D to exactly one open facility i(j) and to connect all open facilities by a Steiner tree T such that the cost \(\sum_{i \in \hat F} f_i + \sum_{j \in D} d_j c_{i(j)j}+M\sum_{e \in T}c_e\) is minimized. We propose a LP-rounding based 8.29 approximation algorithm which improves the previous bound 8.55. We also consider the problem when opening cost of all facilities are equal. In this case we give a 7.0 approximation algorithm.

2012

We consider an online facility location problem where clients arrive over time and their demands have to be served by opening facilities and assigning the clients to opened facilities. When opening a facility we must choose one of K different lease types to use. A lease type k has a certain lease length l k. Opening a facility i using lease type k causes a cost of f k i and ensures that i is open for the next l k time steps. In addition to costs for opening facilities, we have to take connection costs c i j into account when assigning a client j to facility i. We develop and analyze the first online algorithm for this problem that has a time-independent competitive factor. This variant of the online facility location problem was introduced by Nagarajan and Williamson [7] and is strongly related to both the online facility problem by Meyerson [5] and the parking permit problem by Meyerson [6]. Nagarajan and Williamson gave a 3-approximation algorithm for the offline problem and an O(K log n)-competitive algorithm for the online variant. Here, n denotes the total number of clients arriving over time. We extend their result by removing the dependency on n (and thereby on the time). In general, our algorithm is O(l max log(l max))-competitive. Here l max denotes the maximum lease length. Moreover, we prove that it is O(log 2 (l max))-competitive for many "natural" cases. Such cases include, for example, situations where the number of clients arriving in each time step does not vary too much, or is non-increasing, or is polynomially bounded in l max .

SN Computer Science

Facility location (FL) is a well-known optimization problem that asks to optimally place facilities so as to serve clients at various locations, requesting a facility service, with minimum possible costs. Many variants of FL have been known, appearing as sub-problems in many applications in computer science, management science, and operations research. Most FL models studied thus far assume that clients need to be served by connecting each to one facility. To overcome facility failures and provide a robust solution, we investigate in this paper FL problems that require each client to be connected to multiple facilities, represented by an additional input parameter. The aim of the algorithm is then to provide a robust service to all clients while minimizing the total connecting and facility purchasing costs. This is known as the Multi-Facility Location problem (MFL) and has been studied in the offline setting, in which the entire input sequence is given to the algorithm at once. In this paper, we study MFL in the online setting, in which client requests are not known in advance but are revealed to the algorithm over time. We refer to it as the online multi-facility location problem (OMFL) and study its metric and non-metric variants. We propose the first online algorithms for these variants and measure their performance using the standard notion of competitive analysis. The latter is a worst-case analysis that compares the cost of the online algorithm to that of the optimal offline algorithm that is assumed to know all demands in advance. We further study OMFL in the leasing setting, in which facilities are leased, rather than bought, for different durations and prices, and each arriving client needs to be connected to multiple facilities leased at the time of its arrival. The aim is to minimize the total connecting and facility leasing costs.

2002

We consider the Connected Facility Location problem. We are given a graph G = (V, E) with costs {c e } on the edges, a set of facilities F ⊆ V , and a set of clients D ⊆ V . Facility i has a facility opening cost f i and client j has d j units of demand. We are also given a parameter M ≥ 1. A solution opens some facilities, say F , assigns each client j to an open facility i(j), and connects the open facilities by a Steiner tree T . The total cost incurred is i∈F f i + j∈D d j c i(j)j + M e∈T c e . We want a solution of minimum cost.

Lecture Notes in Computer Science, 2011

In the classical facility location problem we are given a set of facilities, with associated opening costs, and a set of clients. The goal is to open a subset of facilities, and to connect each client to the closest open facility, so that the total connection and opening cost is minimized. In some applications, however, open facilities need to be connected via an infrastructure. Furthermore, connecting two facilities among them is typically more expensive than connecting a client to a facility (for a given path length). This scenario motivated the study of the connected facility location problem (CFL). Here we are also given a parameter M ≥ 1. A feasible solution consists of a subset of open facilities and a Steiner tree connecting them. The cost of the solution is now the opening cost, plus the connection cost, plus M times the cost of the Steiner tree. In this paper we investigate the approximability of CFL and related problems. More precisely, we achieve the following results:

2020

We consider a natural extension to the metric uncapacitated Facility Location Problem (FLP) in which requests ask for different commodities out of a finite set S of commodities. Ravi and Sinha (SODA 2004) introduced the model as the Multi-Commodity Facility Location Problem (MFLP) and considered it an offline optimization problem. The model itself is similar to the FLP: i.e., requests are located at points of a finite metric space and the task of an algorithm is to construct facilities and assign requests to facilities while minimizing the construction cost and the sum over all assignment distances. In addition, requests and facilities are heterogeneous; they request or offer multiple commodities out of the set S. A request has to be connected to a set of facilities jointly offering the commodities demanded by it. In comparison to the FLP, an algorithm has to decide not only if and where to place facilities, but also which commodities to offer at each. To the best of our knowledge we are the first to study the problem in its online variant in which requests, their positions and their commodities are not known beforehand but revealed over time. We present results regarding the competitive ratio. On the one hand, we show that heterogeneity influences the competitive ratio by developing a lower bound on the competitive ratio for any randomized online algorithm of Ω(|S| + log n log log n) that already holds for simple line metrics. Here, n is the number of requests. On the other side, we establish a deterministic O(|S| • log n)-competitive algorithm and a randomized O(|S| • log n log log n)-competitive algorithm for the problem. Further, we show that when considering a more special class of cost functions for the construction cost of a facility, the competitive ratio decreases given by our deterministic algorithm depending on the function.

2008

We study the problem of leasing facilities over time, following the general infrastructure leasing problem framework introduced by Anthony and Gupta [1]. If there are K different lease types, Anthony and Gupta give an O(K)-approximation algorithm for the problem. We are able to improve this to a 3-approximation algorithm by using a variant of the primal-dual facility location algorithm of Jain and Vazirani [5]. We also consider the online version of the facility leasing problem, in which the clients to be served arrive over time and are not known in advance. This problem generalizes both the online facility location problem (introduced by Meyerson [6]) and the parking permit problem (also introduced by Meyerson [7]). We give a deterministic algorithm for the problem that is O(K log n)-competitive. To achieve our result, we modify an O(log n)-competitive algorithm of Fotakis [2] for the online facility location problem.

WALCOM: Algorithms and Computation, 2018

We consider the online facility assignment problem, with a set of facilities F of equal capacity l in metric space and customers arriving one by one in an online manner. We must assign customer ci to facility fj before the next customer ci+1 arrives. The cost of this assignment is the distance between ci and fj. The total number of customers is at most |F |l and each customer must be assigned to a facility. The objective is to minimize the sum of all assignment costs. We first consider the case where facilities are placed on a line so that the distance between adjacent facilities is the same and customers appear anywhere on the line. We describe a greedy algorithm with competitive ratio 4|F | and another one with competitive ratio |F |. Finally, we consider a variant in which the facilities are placed on the vertices of a graph and two algorithms in that setting.

Journal of Applied and Industrial Mathematics, 2011

We consider the competitive facility location problem in which two competing sides (the Leader and the Follower) open in succession their facilities, and each consumer chooses one of the open facilities basing on its own preferences. The problem amounts to choosing the Leader's facility locations so that to obtain maximal profit taking into account the subsequent facility location by the Follower who also aims to obtain maximal profit. We state the problem as a two-level integer programming problem. A method is proposed for calculating an upper bound for the maximal profit of the Leader. The corresponding algorithm amounts to constructing the classical maximum facility location problem and finding an optimal solution to it. Simultaneously with calculating an upper bound we construct an initial approximate solution to the competitive facility location problem. We propose some local search algorithms for improving the initial approximate solutions. We include the results of some simulations with the proposed algorithms, which enable us to estimate the precision of the resulting approximate solutions and give a comparative estimate for the quality of the algorithms under consideration for constructing the approximate solutions to the problem.