A new approximation algorithm for the multilevel facility location problem (original) (raw)
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A 3-approximation algorithm for the -level uncapacitated facility location problem
Information Processing Letters, 1999
In the k-level uncapacitated facility location problem, we have a set of demand points where clients are located. The demand of each client is known. Facilities have to be located at given sites in order to service the clients, and each client is to be serviced by a sequence of k different facilities, each of which belongs to a distinct level. There are no capacity restrictions on the facilities. There is a positive fixed cost of setting up a facility, and a per unit cost of shipping goods between each pair of locations. We assume that these distances are all nonnegative and satisfy the triangle inequality. The problem is to find an assignment of each client to a sequence of k facilities, one at each level, so that the demand of each client is satisfied, for which the sum of the setup costs and the service costs is minimized.
Improved Approximation Algorithms for the Uncapacitated Facility Location Problem
SIAM Journal on Computing, 2003
We consider the uncapacitated facility location problem. In this problem, there is a set of locations at which facilities can be built; a fixed cost f i is incurred if a facility is opened at location i. Furthermore, there is a set of demand locations to be serviced by the opened facilities; if the demand location j is assigned to a facility at location i, then there is an associated service cost proportional to the distance between i and j, c ij . The objective is to determine which facilities to open and an assignment of demand points to the opened facilities, so as to minimize the total cost. We assume that the distance function c is symmetric and satisfies the triangle inequality. For this problem we obtain a (1 + 2/e)-approximation algorithm, where 1 + 2/e ≈ 1.736, which is a significant improvement on the previously known approximation guarantees.
Improved approximation algorithms for a capacitated facility location problem
Proceedings of the Tenth Annual Acm Siam Symposium on Discrete Algorithms, 1999
In a recent surprising result, Korupolu, Plaxton, and Rajaraman [10,11] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8 + of the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces a solution which is within a factor of 6(1 +) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron. Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn any α-approximation algorithm for this variant into one which, at an additional cost of twice the optimum of the standard CFLP, opens at most one additional copy of any facility. This allows us to transform a recent 3-approximation algorithm of Chudak and Shmoys [5] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than five times the value of the standard CFLP.
An 0.828-approximation algorithm for the uncapacitated facility location problem
Discrete Applied Mathematics, 1999
The uncapacitated facility location problem in the following formulation is considered: max S⊆I Z(S) = j∈J max i∈S b ij − i∈S c i , where I and J are finite sets, and b ij , c i ≥ 0 are rational numbers. Let Z * denote the optimal value of the problem and Z R = j∈J min i∈I b ij − i∈I c i. Cornuejols, Fisher and Nemhauser (1977) prove that for the problem with the additional cardinality constraint |S| ≤ K, a simple greedy algorithm finds a feasible solution S such that (Z(S)−Z R)/(Z * −Z R) ≥ 1−e −1 ≈ 0.632. We suggest a polynomial-time approximation algorithm for the unconstrained version of the problem, based on the idea of randomized rounding due to Goemans and Williamson (1994). It is proved that the algorithm delivers a solution S such that (Z(S) − Z R)/(Z * − Z R) ≥ 2(√ 2 − 1) ≈ 0.828. We also show that there exists ε > 0 such that it is N P-hard to find an approximate solution S with (Z(S) − Z R)/(Z * − Z R) ≥ 1 − ε.
Applied Mathematical Modelling, 2014
This paper considers the multi level uncapacitated facility location problem (MLUFLP). A new mixed integer linear programming (MILP) formulation is presented and validity of this formulation is given. Experimental results are performed on instances known from literature. The results achieved by CPLEX and Gurobi solvers, based on the proposed MILP formulation, are compared to the results obtained by the same solvers on the already known formulations. The results show that CPLEX and Gurobi can optimally solve all small and medium sized instances and even some large-scale instances using the new formulation.
Approximate algorithms for the competitive facility location problem
Journal of Applied and Industrial Mathematics, 2011
We consider the competitive facility location problem in which two competing sides (the Leader and the Follower) open in succession their facilities, and each consumer chooses one of the open facilities basing on its own preferences. The problem amounts to choosing the Leader's facility locations so that to obtain maximal profit taking into account the subsequent facility location by the Follower who also aims to obtain maximal profit. We state the problem as a two-level integer programming problem. A method is proposed for calculating an upper bound for the maximal profit of the Leader. The corresponding algorithm amounts to constructing the classical maximum facility location problem and finding an optimal solution to it. Simultaneously with calculating an upper bound we construct an initial approximate solution to the competitive facility location problem. We propose some local search algorithms for improving the initial approximate solutions. We include the results of some simulations with the proposed algorithms, which enable us to estimate the precision of the resulting approximate solutions and give a comparative estimate for the quality of the algorithms under consideration for constructing the approximate solutions to the problem.
Improved Approximation Algorithm for Connected Facility Location Problems
2007
We study the Connected Facility Location problems. We are given a connected graph G = (V, E) with non-negative edge cost c e for each edge e ∈ E, a set of clients D ⊆ V such that each client j ∈ D has positive demand d j and a set of facilities F ⊆ V each has non-negative opening cost f i and capacity to serve all client demands. The objective is to open a subset of facilities, say \(\hat F\) , to assign each client j ∈ D to exactly one open facility i(j) and to connect all open facilities by a Steiner tree T such that the cost \(\sum_{i \in \hat F} f_i + \sum_{j \in D} d_j c_{i(j)j}+M\sum_{e \in T}c_e\) is minimized. We propose a LP-rounding based 8.29 approximation algorithm which improves the previous bound 8.55. We also consider the problem when opening cost of all facilities are equal. In this case we give a 7.0 approximation algorithm.
Improved approximation guarantees for lower-bounded facility location problem
2010
We consider the lower-bounded facility location (LBFL) problem (also sometimes called load-balanced facility location), which is a generalization of uncapacitated facility location (UFL), where each open facility is required to serve a certain minimum amount of demand. More formally, an instance I of LBFL is specified by a set F of facilities with facility-opening costs {f i }, a set D of clients, and connection costs {c ij } specifying the cost of assigning a client j to a facility i, where the c ij s form a metric. A feasible solution specifies a subset F of facilities to open, and assigns each client j to an open facility i(j) ∈ F so that each open facility serves at least M clients, where M is an input parameter. The cost of such a solution is i∈F f i + j c i(j)j , and the goal is to find a feasible solution of minimum cost. The current best approximation ratio for LBFL is 448 [18]. We substantially advance the state-of-theart for LBFL by devising an approximation algorithm for LBFL that achieves a significantly-improved approximation guarantee of 82.6. Our improvement comes from a variety of ideas in algorithm design and analysis, which also yield new insights into LBFL. Our chief algorithmic novelty is to present an improved method for solving a more-structured LBFL instance obtained from I via a bicriteria approximation algorithm for LBFL, wherein all clients are aggregated at a subset F ′ of facilities, each having at least αM co-located clients (for some α ∈ [0, 1]). One of our key insights is that one can reduce the resulting LBFL instance, denoted I 2 (α), to a problem we introduce, called capacity-discounted UFL (CDUFL). CDUFL is a special case of capacitated facility location (CFL) where facilities are either uncapacitated, or have finite capacity and zero opening costs. Circumventing the difficulty that CDUFL inherits the intractability of CFL with respect to LP-based approximation guarantees, we give a simple local-search algorithm for CDUFL based on add, delete, and swap moves that achieves the same approximation ratio (of 1 + √ 2) as the corresponding local-search algorithm for UFL. In contrast, the algorithm in [18] proceeds by reducing I 2 (α) to CFL, whose current-best approximation ratio is worse than that of our local-search algorithm for CDUFL, and this is one of the reasons behind our algorithm's improved approximation ratio. Another new ingredient of our LBFL-algorithm and analysis is a subtly different method for constructing a bicriteria solution for I (and hence, I 2 (α)), combined with the more significant change that we now choose a random α from a suitable distribution. This leads to a surprising degree of improvement in the approximation factor, which is reminiscent of the mileage provided by random α-points in scheduling problems.
An Approximation Algorithm for the k-Level Uncapacitated Facility Location Problem with Penalties
2009
The k-level Uncapacitated Facility Location (UFL) problem is a generalization of the UFL and the k-median problems. A significant shortcoming of the classical UFL problem is that often a few very distant customers, known as outliers, can leave an undesirable effect on the final solution. This deficiency is considered in a new variant called UFL with outliers, in which, in contrast to the other problems that need all of the customers to be serviced, there is no need to service the entire set of customers.