Approximation Algorithms for Fault Tolerant Facility Allocation

Abstract

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.