Unconstrained and Constrained Fault-Tolerant Resource Allocation

First, we study the Unconstrained Fault-Tolerant Resource Allocation (UFTRA) problem (a.k.a. FTFA problem in \cite{shihongftfa}). In the problem, we are given a set of sites equipped with an unconstrained number of facilities as resources, and a set of clients with set $\mathcal{R}$ as corresponding connection requirements, where every facility belonging to the same site has an identical opening (operating) cost and every client-facility pair has a connection cost. The objective is to allocate facilities from sites to satisfy $\mathcal{R}$ at a minimum total cost. Next, we introduce the Constrained Fault-Tolerant Resource Allocation (CFTRA) problem. It differs from UFTRA in that the number of resources available at each site $i$ is limited by $R_{i}$. Both problems are practical extensions of the classical Fault-Tolerant Facility Location (FTFL) problem \cite{Jain00FTFL}. For instance, their solutions provide optimal resource allocation (w.r.t. enterprises) and leasing (w.r.t. clients) strategies for the contemporary cloud platforms. In this paper, we consider the metric version of the problems. For UFTRA with uniform $\mathcal{R}$, we present a star-greedy algorithm. The algorithm achieves the approximation ratio of 1.5186 after combining with the cost scaling and greedy augmentation techniques similar to \cite{Charikar051.7281.853,Mahdian021.52}, which significantly improves the result of \cite{shihongftfa} using a phase-greedy algorithm. We also study the capacitated extension of UFTRA and give a factor of 2.89. For CFTRA with uniform $\mathcal{R}$, we slightly modify the algorithm to achieve 1.5186-approximation. For a more general version of CFTRA, we show that it is reducible to FTFL using linear programming

Approximation algorithms for fault tolerant facility allocation

Given nf 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(mRlogm) and O(Rn3), respectively, where R is the maximum number of facilities required by any city, m = nfnc, and n = max{ nf, nc}. 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. ©2013 Society for Industrial and Applied Mathematics.Hong Shen and Shihong X

LP-based approximation algorithms for reliable resource allocation

We initiate the study of the reliable resource allocation (RRA) problem. In this problem, we are given a set of sites ℱ each with an unconstrained number of facilities as resources. Every facility at site i ∈ ℱ has an opening cost and a service reliability pi. There is also a set of clients \u1d49e to be allocated to facilities. Every client j ∈ \u1d49e accesses a facility at i with a connection cost and reliability lij. In addition, every client j has a minimum reliability requirement (MRR) rj for accessing facilities. The objective of the problem is to decide the number of facilities to open at each site and connect these facilities to clients such that all clients’ MRRs are satisfied at a minimum total cost. The unconstrained fault-tolerant resource allocation problem studied in Liao and Shen [(2011) Unconstrained and Constrained Fault-Tolerant Resource Allocation. Proceedings of the 17th Annual International Conference on Computing and Combinatorics (COCOON), Dallas, Texas, USA, August 14–16, pp. 555–566. Springer, Berlin] is a special case of RRA. Both of these resource allocation problems are derived from the classical facility location theory. In this paper, for solving the general RRA problem, we develop two equivalent primal-dual algorithms where the second one is an acceleration of the first and runs in quasi-quadratic time. In the algorithm's ratio analysis, we first obtain a constant approximation factor of 2+2√2 and then a reduced ratio of 3.722 using a factor revealing program, when lij's are uniform on i (partially uniform) and rj's are uniform above the threshold reliability that a single access to a facility is able to provide. The analysis further elaborates and generalizes the inverse dual-fitting technique introduced in Xu and Shen [(2009) The Fault-Tolerant Facility Allocation Problem. Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC), Honolulu, HI, USA, December 16–18, pp. 689–698. Springer, Berlin]. Moreover, we formalize this technique for analyzing the minimum set cover problem. For a special case of RRA, where all rj's and lij's are uniform, we derive its approximation ratio through a novel reduction to the uncapacitated facility location problem. The reduction demonstrates some useful and generic linear programming techniques