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 R_i facilities with opening cost f_i. Each client j requires an allocation of r_j open facilities and connecting j to any facility at site i incurs a connection cost c_ij. 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 R_i, whereas FTFL sets R_i=1. These problems are said to be uniform if all r_j'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.