Fault-tolerant Facility Location Research Articles

Constant approximation for fault-tolerant median problems via iterative rounding

In this paper, we study the fault-tolerant matroid median and fault-tolerant knapsack median problems. These two problems generalize many fundamental clustering and facility location problems, such as uniform fault-tolerant k-median, uniform fault-tolerant facility location, matroid median, knapsack median, etc. We present a versatile iterative rounding framework and obtain a unifying constant-factor approximation algorithm.

Mixed fault tolerance in server assignment: Combining reinforcement and backup

We study the mixed approach to fault tolerance in the general context of server assignment in networks. The approach is based on mixing two different existing strategies, namely, reinforcement and backup. The former strategy protects clients by reinforcing the servers assigned to them and making them fault-resistant, possibly at a high cost, while the latter protects clients by assigning to them alternate low price backup servers that can replace their primary servers in case those fail. Applying the mixed approach to fault tolerance gives rise to new fault tolerant variations of known server assignment problems. We introduce several NP-hard problems of this type, including the mixed fault-tolerant dominating set problem, the mixed fault-tolerant centers problem, and the mixed fault-tolerant facility location problem, and present polynomial time approximation algorithms for them, demonstrating the viability of the mixed strategy for server assignment problems.

Approximation algorithms for the fault-tolerant facility location problem with penalties

We study the fault-tolerant facility location problem with penalties (FTFLPWP). We present an LP-rounding 4-approximation algorithm. Then we apply the randomized rounding technique to improve the approximation to 3.16, which is further improved to 2.408 by the greedy augmentation technique.

LP-rounding approximation algorithms for two-stage stochastic fault-tolerant facility location problem

• The two-stage stochastic version for the fault-tolerant facility location problem is introduced. • Presenting a deterministic LP-rounding 5-approximation algorithm. • The ratio is further improved to 3.8617 by using randomization technique. In this paper, we study the weighted two-stage stochastic fault-tolerant facility location problem. We present a deterministic LP-rounding 5-approximation algorithm by exploiting both of its stochastic and fault-tolerant structures. We further offer an improved randomized LP-rounding 3.8617-approximation algorithm along with the corresponding derandomized version with the same approximation ratio.

A Constant Factor Approximation Algorithm for Fault-Tolerant k -Median

In this article, we consider the fault-tolerant k -median problem and give the first constant factor approximation algorithm for it. In the fault-tolerant generalization of the classical k -median problem, each client j needs to be assigned to at least r j ⩾ 1 distinct open facilities. The service cost of j is the sum of its distances to the r j facilities, and the k -median constraint restricts the number of open facilities to at most k . Previously, a constant factor was known only for the special case when all r j s are the same, and alogarithmic approximation ratio was known for the general case. In addition, we present the first polynomial time algorithm for the fault-tolerant k -median problem on a path or an HST by showing that the corresponding LP always has an integral optimal solution. We also consider the fault-tolerant facility location problem, in which the service cost of j can be a weighted sum of its distance to the r j facilities. We give a simple constant factor approximation algorithm, generalizing several previous results that work only for nonincreasing weight vectors.

Improved approximation algorithms for constrained fault-tolerant resource allocation

In Constrained Fault-Tolerant Resource Allocation (FTRA) problem, we are given a set of sites containing facilities as resources and a set of clients accessing these resources. Each site i can open at most Ri facilities with opening cost fi. Each client j requires an allocation of rj open facilities and connecting j to any facility at site i incurs a connection cost cij. The goal is to minimize the total cost of this resource allocation scenario. FTRA generalizes the Unconstrained Fault-Tolerant Resource Allocation (FTRA∞) [1] and the classical Fault-Tolerant Facility Location (FTFL) [2] problems: for every site i, FTRA∞ does not have the constraint Ri, whereas FTFL sets Ri=1. These problems are said to be uniform if all rj's are the same, and general otherwise. For the general metric FTRA, we first give an LP-rounding algorithm achieving an approximation ratio of 4. Then we show the problem reduces to FTFL, implying the ratio of 1.7245 from [3]. For the uniform FTRA, we provide a 1.52-approximation primal–dual algorithm in O(n4) time, where n is the total number of sites and clients.

An Approximation Algorithm for the Stochastic Fault-Tolerant Facility Location Problem

In this paper, we study a stochastic version of the fault-tolerant facility location problem. By exploiting the stochastic structure, we propose a 5-approximation algorithm which uses the LP-rounding technique based on the revised optimal solution to the linear programming relaxation of the stochastic fault-tolerant facility location problem.

Distributed Approximation Algorithms for Facility Location Problems

In this paper, some distributed approximation algorithms for uncapacitated facility location problem and fault-tolerant facility location problem are summarized, respectively. The worst-case run time of the distributed version of the uncapacitated facility location problem is ( ) O n , where n is the number of network nodes. In addition, the approximation approach for the fault-tolerant facility location problem is also implemented in a distributed and asynchronous manner within ( ) O n rounds of communication.

Approximation Algorithms for Fault Tolerant Facility Allocation

Given $n_{f}$ sites, each equipped with one facility, and $n_{c}$ cities, fault tolerant facility location (FTFL) [K. Jain and V. V. Vazirani, APPROX '$00$: Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization, Spinger, New York, 2000, pp. 177--183] requires computing a minimum-cost connection scheme such that each city connects to a specified number of facilities. When each city connects to exactly one facility, FTFL becomes the classical uncapacitated facility location problem (UFL) that is well-known NP hard. The current best solution to FTFL admits an approximation ratio 1.7245 due to Byrka, Srinivasan, and Swamy applying the dependent rounding technique announced recently [Proceedings of IPCO, 2010, pp. 244--257], which improves the ratio 2.076 obtained by Swamy and Shmoys based on LP rounding [ACM Trans. Algorithms, 4 (2008), pp. 1--27]. In this paper, we study a variant of the FTFL problem, namely, fault tolerant facility allocation (FTFA), as another generalization of UFL by allowing each site to hold multiple facilities and show that we can obtain better solutions for this problem. We first give two algorithms with 1.81 and 1.61 approximation ratios in time complexity $O(mR\log m)$ and $O(Rn^{3})$, respectively, where $R$ is the maximum number of facilities required by any city, $m=n_{f}n_{c}$, and $n=\max\{n_{f},n_{c}\}$. Instead of applying the dual-fitting technique that reduces the dual problem's solution to fit the original problem as used in the literature [K. Jain et al., Journal of the ACM, 50 (2003), pp. 795--824; K. Jain, M. Mahdian, and A. Saberi, STOC'02: Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, New York, 2002, pp. 731--740; A. Saberi et al., Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques, Springer, New York, 2001, pp. 127--137], we propose a method called inverse dual-fitting that alters the original problem to fit the dual solution and show that this method is more effective for obtaining solutions of multifactor approximation. We show that applying inverse dual-fitting and factor-revealing techniques our second algorithm is also (1.11,1.78)- and (1,2)-approximation simultaneously. These results can be further used to achieve solutions of 1.52-approximation to FTFA and 4-approximation to the fault tolerant k-facility allocation problem in which the total number of facilities is bounded by $k$. These are currently the best bifactor and single-factor approximation ratios for the problems concerned.

Approximating soft-capacitated facility location problem with uncertainty

In this paper we devise the stochastic and robust approaches to study the soft-capacitated facility location problem with uncertainty. We first present a new stochastic soft-capacitated model called The 2-Stage Soft Capacitated Facility Location Problem and solve it via an approximation algorithm by reducing it to linear-cost version of 2-stage facility location problem and dynamic facility location problem. We then present a novel robust model of soft-capacitated facility location, The Robust Soft Capacitated Facility Location Problem. To solve it, we improve the approximation algorithm proposed by Byrka et al. (LP-rounding algorithms for facility-location problems. CoRR, 2010a) for RFTFL and then treat it similarly as in the stochastic case. The improvement results in an approximation factor of $$\alpha + 4$$ for the robust fault-tolerant facility location problem, which is best so far.

Fault-tolerant concave facility location problem with uniform requirements

In this paper, we consider the fault-tolerant concave facility location problem (FTCFL) with uniform requirements. By investigating the structure of the FTCFL, we obtain a modified dual-fitting bifactor approximation algorithm. Combining the scaling and greedy argumentation technique, the approximation factor is proved to be 1.52.

Improved approximation algorithms for the robust fault-tolerant facility location problem

We consider the robust α fault-tolerant facility location problem (α-RFLP), recently introduced by Chechik and Peleg (2010) [6]. We present an improved approximation algorithm with ratio 5.236 for the 1-RFLP comparing to 6.5 by Chechik and Pelegʼs. For the general α-RFLP (fixed α⩾2), the same algorithm with a different subroutine tailored for α⩾2 provides an improved approximation ratio 1.005+6.015α comparing to 1.5+7.5α by Chechik and Pelegʼs. The key component of our algorithm is the resolution of an auxiliary facility location problem (FLP) by a variant of the LP-rounding technique of Byrka and Aardal (2010) [2] to estimate the total weighted facility open cost and shipping cost.

Fault-tolerant facility location

We consider a fault-tolerant generalization of the classical uncapacitated facility location problem, where each client j has a requirement that r j distinct facilities serve it, instead of just one. We give a 2.076-approximation algorithm for this problem using LP rounding, which is currently the best-known performance guarantee. Our algorithm exploits primal and dual complementary slackness conditions and is based on clustered randomized rounding . A technical difficulty that we overcome is the presence of terms with negative coefficients in the dual objective function, which makes it difficult to bound the cost in terms of dual variables. For the case where all requirements are the same, we give a primal-dual 1.52-approximation algorithm. We also consider a fault-tolerant version of the k -median problem. In the metric k -median problem, we are given n points in a metric space. We must select k of these to be centers, and then assign each input point j to the selected center that is closest to it. In the fault-tolerant version we want j to be assigned to r j distinct centers. The goal is to select the k centers so as to minimize the sum of assignment costs. The primal-dual algorithm for fault-tolerant facility location with uniform requirements also yields a 4-approximation algorithm for the fault-tolerant k -median problem for this case. This the first constant-factor approximation algorithm for the uniform requirements case.

A constant factor approximation algorithm for the fault-tolerant facility location problem

We consider a generalization of the classical facility location problem, where we require the solution to be fault-tolerant. In this generalization, every demand point j must be served by r j facilities instead of just one. The facilities other than the closest one are “backup” facilities for that demand, and any such facility will be used only if all closer facilities (or the links to them) fail. Hence, for any demand point, we can assign nonincreasing weights to the routing costs to farther facilities. The cost of assignment for demand j is the weighted linear combination of the assignment costs to its r j closest open facilities. We wish to minimize the sum of the cost of opening the facilities and the assignment cost of each demand j. We obtain a factor 4 approximation to this problem through the application of various rounding techniques to the linear relaxation of an integer program formulation. We further improve the approximation ratio to 3.16 using randomization and to 2.41 using greedy local-search type techniques.