A new approximation algorithm for the multilevel facility location problem

In this paper we propose a new integer programming formulation for the multi-level facility location problem and a novel 3-approximation algorithm based on LP rounding. The linear program we are using has a polynomial number of variables and constraints, being thus more efficient than the one commonly used in the approximation algorithms for this type of problems

An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem

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 {\em ($\lambda_f$,$\lambda_c$)-approximation algorithm} if the solution it produces has total cost at most $\lambda_f \cdot F^* + \lambda_c \cdot C^*$, 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 $(\gamma_f, 1+2e^{-\gamma_f})$ 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.Comment: A journal versio

Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixed-charge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our technique, we obtain a new 1.61-approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lot-sizing problem with general concave ordering costs, and a 4-approximation algorithm for the joint replenishment problem with general concave individual ordering costs

A simple dual ascent algorithm for the multilevel facility location problem

We present a simple dual ascent method for the multilevel facility location problem which finds a solution within $6$ times the optimum for the uncapacitated case and within $12$ times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique. \u

Robust Fault Tolerant uncapacitated facility location

In the uncapacitated facility location problem, given a graph, a set of demands and opening costs, it is required to find a set of facilities R, so as to minimize the sum of the cost of opening the facilities in R and the cost of assigning all node demands to open facilities. This paper concerns the robust fault-tolerant version of the uncapacitated facility location problem (RFTFL). In this problem, one or more facilities might fail, and each demand should be supplied by the closest open facility that did not fail. It is required to find a set of facilities R, so as to minimize the sum of the cost of opening the facilities in R and the cost of assigning all node demands to open facilities that did not fail, after the failure of up to \alpha facilities. We present a polynomial time algorithm that yields a 6.5-approximation for this problem with at most one failure and a 1.5 + 7.5\alpha-approximation for the problem with at most \alpha > 1 failures. We also show that the RFTFL problem is NP-hard even on trees, and even in the case of a single failure