A new approximation algorithm for the multilevel facility location problem (original) (raw)
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 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.
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.
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.
A Comparison of Exact and Heuristic Methods for a Facility Location Problem
International journal of simulation: systems, science & technology
We formulate a facility location problem where the demand of any single client must be allocated to a single facility and a prize is obtained by allocating the demand of a client to a certain facility, i.e. a prize-based variant of the Single Source Capacitated Facility Location Problem. For this problem we pursue both an exact approach through Integer Linear Programming and a heuristic approach based on a local search algorithm. We compare both approaches by considering 500+ instances. The heuristic approach allows to obtain a reduction of the computational time by a factor larger than 10 in 92% of instances and 100 in 64% of instances. The time reduction is obtained with a small sacrifice in the value of the objective function that is achieved, smaller than 10% in nearly 70% of cases.
Approximation Algorithms for Metric Facility Location Problems
Siam Journal on Computing, 2006
In this paper we present a 1.52-approximation algorithm for the metric uncapacitated facility location problem, and a 2-approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for the corresponding problem, and our soft-capacitated facility location algorithm achieves the integrality gap of the standard LP relaxation of the problem. Furthermore, we will show, using a result of Thorup, that our algorithms can be implemented in quasi-linear time.
Facility location with adjacent units: a simple approximation scheme
International Journal of Information and Operations Management Education, 2013
In this paper, we provide a simple approximation scheme for the optimal objective value for a particular type of location problem. Typically, such problems are solved using the classic set covering formulation. Such a formulation automatically requires data for the constraint matrix and can get too large to implement or too difficult to solve to optimality. The scheme presented in this paper has minimal need for such data. Based on a simple count and with some basic and realistic assumptions about the geometry of the problem, we provide an algebraic formula that gives a close approximation to the optimal objective function value. Our formula can be easily implemented in a spreadsheet or hand-held calculator making it an effective planning tool for practice and also a good pedagogical aid. We illustrate by applying it to a location problem involving individual states in the continental US and collectively to the entire country.
Approximation algorithms for a facility location problem with service capacities
ACM Transactions on Algorithms, 2008
In this article we focus on approximation algorithms for facility location problems with subadditive costs. As examples of such problems, we present three facility location problems with stochastic demand and exponential servers, respectively inventory. We present a (1 + , 1)-reduction of the facility location problem with subadditive costs to the soft capacitated facility location problem, which implies the existence of a 2(1 + )-approximation algorithm. For a special subclass of subadditive functions, we obtain a 2-approximation algorithm by reduction to the linear cost facility location problem.
Approximation algorithms for the capacitated k-facility location problems
2013
In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated kkk-facility location problem. Then, we show that the capacitated kkk-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated kkk-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening cost.
Improved Approximation Guarantees for Lower-Bounded Facility Location
Approximation and Online Algorithms, 2013
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 Optimal Bifactor Approximation Algorithm for the Metric Uncapacitated Facility Location Problem
Siam Journal on Computing, 2010
We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location (UFL) problem, 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 (λf ,λc)-approximation algorithm if the solution it produces has total cost at most
Computers & Operations Research, 2017
We consider the two-level uncapacitated facility location problem with single assignment constraints (TUFLP-S), an extension of the uncapacitated facility location problem. We present six mixed-integer programming models for the TUFLP-S based on reformulation techniques and on the relaxation of the integrality of some of the variables associated with location decisions. We compare the models by carrying out extensive computational experiments on large, hard, artificial instances, as well as on instances derived from an industrial application in freight transportation.
Approximation algorithms for hard capacitated k-facility location problems
European Journal of Operational Research, 2015
We study the capacitated k-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number k. It costs f i to open facility i, and c ij for facility i to serve one unit of demand from client j. The objective is to open at most k facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs. In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated k-facility location problem. Then, we show that the capacitated k-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated k-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening cost.
Brazilian Journal of Operations & Production Management, 2017
The objective in terms of the facility location problem with limited distances is to minimize the sum of distance functions from the facility to its clients, but with a limit on each of these distances, from which the corresponding function becomes constant. The problem is applicable in situations where the service provided by the facility is insensitive after given threshold distances. In this paper, we propose a polynomial-time algorithm for the discrete version of the problem with capacity constraints regarding the number of served clients. These constraints are relevant for introducing quality measures in facility location decision processes as well as for justifying the facility creation.