An Optimal Bifactor Approximation Algorithm for the Metric Uncapacitated Facility Location Problem

Abstract

We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463..An algorithm is a (λ f ,λ c )-approximation algorithm if the solution it produces has total cost at most λ f • F * + λ c • C * , where F * and C * are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the (1 + 2/e)-approximation algorithm of Chudak and Shmoys, is a (1.6774,1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve (γ f , 1 + 2e −γ f ) established by Jain, Mahdian and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11,1.7764)-approximation algorithm proposed by Jain et al., and later analyzed by Mahdian et al., we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL.

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

“…Recently, Li [21] offered an improved analysis of the LP rounding algorithm by Byrka and Aardal [4] to obtain the currently best approximation ratio for the classical FLP.…”

“…Following the first constantfactor approximation algorithm by Shmoys et al [24], there was a long list of work on designing improved approximation algorithms for this problem over the years. As a result, four basic schemes have emerged in the design of these approximation algorithms, namely LP-rounding [4,9,21,24,27], primal-dual [16], dual-fitting [15,17,23], and local search [5,13,18]. These four competitive and complementary schemes possess different features.…”

“…Our algorithm and analysis indicate that the latter is substantially harder to approximate than the former because the techniques for the FLPLP, such as that by Byrka and Aardal [4], are not directly applicable to the FLPSP. To overcome this difficulty, our algorithm for the latter will exploit the special structure of the optimal LP relaxation solution of the FLPSP.…”

A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties

“…Moreover, these approaches can be mixed or combined with other skills such as greedy augmentation, cost scaling and dual-fitting [13,15] to yield better approximation algorithms. So far, the best approximation ratio for the UFLP is 1.50 due to Byrka and Aardal [3]. Their algorithm is based on LP-rounding approach integrated with greedy and dual-fitting.…”

“…They presented an (1 + γ )-approximation algorithm where γ is the approximation factor of an LP-based approximation algorithm for the corresponding UFLP, resulting in a 2.50-approximation algorithm for the FLPSP as the best known ratio of LP-based approximation algorithms for the UFLP is 1.50 [3]. However, their algorithm is noncombinatorial because it has to solve the LP-relaxation of an integer program with exponential number of variables.…”