Approximation algorithms for the Fault-Tolerant Facility Placement problem

Abstract

In the Fault-Tolerant Facility Placement problem (FTFP) we are given a set of customers C, a set of sites F , and distances between customers and sites. We assume that the distances satisfy the triangle inequality. Each customer j has a demand r j and each site may open an unbounded number of facilities. The objective is to minimize the total cost of opening facilities and connecting each customer j to r j different open facilities. We present two LP-rounding algorithms for FTFP. The first algorithm achieves an approximation ratio of 4. The second algorithm uses the method of filtering to improve the ratio to 3.16.Published by Elsevier B.V.

Improved approximation algorithms for constrained fault-tolerant resource allocation

“…This problem is closely related to the Unconstrained Fault-Tolerant Resource Allocation (FTRA ∞ ) 1 [1], the classical Fault-Tolerant Facility Location (FTFL) [2] and Uncapacitated Facility Location (UFL) [6] problems. Both FTRA ∞ and FTFL are the special cases of FTRA: R i is unbounded in FTRA ∞ , whereas ∀i ∈ F : R i = 1 in FTFL.…”

“…In addition, Mahdian et al [11] improved that of the JMS algorithm to 1.52 using the standard cost scaling and greedy augmentation techniques. Shmoys et al [6] first gave a filtering based LP-rounding algorithm achieving the constant ratio of 3.16. Later, Chudak and Shmoys [12] came up with the clustered randomized rounding algorithm which further reduces the ratio to 1.736.…”

“…The general case of the problem was also studied by Yan and Chrobak. They first gave a 3.16-approximation LP-rounding algorithm [5], and then obtained the ratio of 1.575 [20] built on the work of [12,13,21,19]. Recently, Rybicki and Byrka [22] gave an elegant asymptotic approximation algorithm (with various better ratios depending on min j r j ) and some improved hardness results.…”

“…However, there are several difficulties. First, despite the similar combinatorial structures of FTRA ∞ and FTRA, the existing LP-rounding algorithms [5,20] for FTRA ∞ cannot be adopted for FTRA. This is because these algorithms produce infeasible solutions that violate the constraint R i in FTRA.…”

“…This is achieved by: 1) constructing some useful properties of the unified algorithm which enable us to directly round the optimal fractional solutions with values that might exceed one while ensuring the feasibility of the rounded solutions and the algorithm correctness; 2) exploiting the primal and dual complementary slackness conditions of the FTRA problem's LP formulation. Then we show FTRA can reduce to FTFL using an instance shrinking technique inspired from the splitting idea of [23] for FTRA ∞ . It implies that these two problems may share the same approximability in weakly polynomial time.…”