Information Processing Letters 112 (2012) 361–364

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Information Processing Letters

www.elsevier.com/locate/ipl

Improved approximation algorithms for the robust fault-tolerant facilitylocation problem

Yu Li a, Dachuan Xu b,∗, Donglei Du c, Naihua Xiu a

a Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, PR Chinab Department of Applied Mathematics, Beijing University of Technology, Beijing 100124, PR Chinac Faculty of Business Administration, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 August 2010Received in revised form 29 January 2012Accepted 6 February 2012Available online 8 February 2012Communicated by B. Doerr

Keywords:Facility locationApproximation algorithmFault tolerance

We consider the robust α fault-tolerant facility location problem (α-RFLP), recently introducedby Chechik and Peleg (2010) [6]. We present an improved approximation algorithm withratio 5.236 for the 1-RFLP comparing to 6.5 by Chechik and Peleg’s. For the general α-RFLP(fixed α � 2), the same algorithm with a different subroutine tailored for α � 2 providesan improved approximation ratio 1.005 + 6.015α comparing to 1.5 + 7.5α by Chechik andPeleg’s. The key component of our algorithm is the resolution of an auxiliary facility locationproblem (FLP) by a variant of the LP-rounding technique of Byrka and Aardal (2010) [2] toestimate the total weighted facility open cost and shipping cost.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The classical facility location problem (FLP) is an exten-sively studied combinational optimization problem fromapproximation algorithm point of view due to its NP-hardness. Both upper and lower bounds on the approxima-tion ratio for the FLP have been proposed and improvedover the years [2,5,8,10,11,13–15,19]. For other variantsof the FLP, we refer to [1,7,16–18,21–24] and referencestherein.

The fault-tolerant facility location problem (FTFLP) extendsthe above FLP by associating with each client a positive in-teger specifying its coverage requirement r j . The task is tofind a minimum-cost solution which opens some facilitiesand connects each client j to r j different open facilities.This extension was introduced by Jain and Vazirani [12]with the first constant approximation ratio 2.408 due toGuha et al. [9], later improved to 2.076 and 1.725, respec-tively by Swamy and Shmoys [20] and Byrka et al. [4].

* Corresponding author.E-mail addresses: liyubjut@gmail.com (Y. Li), xudc@bjut.edu.cn (D. Xu),

ddu@unb.ca (D. Du), nhxiu@center.njtu.edu.cn (N. Xiu).

0020-0190/$ – see front matter © 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.ipl.2012.02.004

The main focus of this work is the robust α fault-tolerantfacility location problem (α-RFLP), recently introduced byChechik and Peleg [6]. In this problem, at most α facilitiesmight fail, and each client should be supplied by the clos-est open facility that did not fail. The problem is to choosea set of facilities R , so as to minimize the sum of the costof opening the facilities in R and the cost of assigning allclients to the remaining open facilities after the failure ofup to α facilities. In order to design improved approxima-tion algorithm for the α-RFLP, we propose a modified LP-rounding algorithm for an auxiliary uncapacitated facilitylocation problem by adapting the LP-rounding techniqueof Byrka and Aardal [2] to estimate the total weighted fa-cility open cost and shipping cost rather than the simpleaddition of these two costs. Together with the techniquein [6], we give an improved 5.236-approximation algo-rithm for the 1-RFLP comparing to the 6.5-approximationby Chechik and Peleg [6]. The same algorithm with onedifferent subroutine tailored for α � 2 provides an im-proved approximation ratio 1.005 + 6.015α comparing to1.5 + 7.5α by Chechik and Peleg’s. During the preparationof this paper, there is an independent work for the sameproblem with an approximation ratio of 5 + 4/α + α [3],which is better than the results presented here for all

362 Y. Li et al. / Information Processing Letters 112 (2012) 361–364

α � 2. However, this algorithm needs to solve an LP withO (nα+2) decision variables while our algorithm only solvesan LP with O (n2) variables, where n denotes the size of theα-RFLP.

The rest of this paper is organized as follows. In Sec-tion 2, we introduce some notations with respect to theFLP and its variant. In Section 3, we present an LP-rounding based algorithm for the FLP. In Section 4, weoffer improved approximation algorithms for both the 1-RFLP and the α-RFLP (α � 2). Some concluding remarksare given in Section 5.

2. Preliminaries

The FLP can be formally defined as follows. Considera set V of clients. All the nodes in V can also serve asfacilities ready to be opened. For every facility r ∈ V andevery client u ∈ V , there is a nonnegative opening cost fr ,a demand du , and a connection cost cur . We assume thatthe connection cost between clients and facilities is met-ric (i.e., satisfying the triangle inequality). We denote theshipping cost of assigning the demand du of a client u toan open facility r as Sur := ducur . The shipping cost froma node u to a set of open facilities R is Su,R := min{Sur |r ∈ R} = ducu,R , where cu,R := minr∈R cur . The problem isto open a subset R ⊆ V of facilities such that each clientu ∈ V is assigned to an open facility with the objective tominimize the total cost

CFLP(R) := F (R) + S(R),

where

F (R) :=∑r∈R

fr and S(R) :=∑u∈V

Su,R =∑u∈V

ducu,R .

The α-RFLP is a variant of the standard FLP which canbe described as follows. When we open a subset of facili-ties, at most α of them can be closed, resulting in clientsbeing supplied by the closest of the remaining open fa-cilities. This problem is to search for a subset R ⊆ V thatminimizes the cost function

Cα-RFLP(R) := F (R) + maxR ′⊆R, |R ′|�α

{S(

R \ R ′)}.To design algorithms for the α-RFLP, we also consider

another variant of the FLP introduced by Chechik and Pe-leg [6], namely the α-concentrated backup problem (α-CBP).In the α-CBP, there is a given set R1 ⊆ V and the nodes inthe set R1 act as both clients and facilities without opencost. All other nodes v /∈ R1 have zero demands. Assum-ing that there is a set of facilities F (|F | � α) which couldbe closed, each client r ∈ R1 is required to be assignedto an open backup facility v ∈ V \ {F }. For a set of nodesR2 ⊆ V \ R1, define the α backup cost

Cα-B(R1, R2) := max|F |�α

{ ∑r∈F∩R1

Sr,R1∪R2\F

}.

The goal of α-CBP is to look for a set R2 ⊆ V \ R1 whichminimizes the total cost

Cα-CBP(R1, R2) = F (R2) + Cα-B(R1, R2).

For the α-CBP, there is a slightly better approximationalgorithm for α = 1 compared to α � 2 due to Chechik andPeleg [6], summarized below for later usage.

Lemma 2.1. (See [6].) For the α-CBP, (1) there exits a 2-ap-proximation algorithm (denoted as ALG1-CBP) when α = 1, and(2) there exists a 3α-approximation algorithm (denoted asALGα-CBP) when α � 2.

3. A modified LP-rounding algorithm for the FLP

3.1. Formulation

Before providing our algorithm for the α-RFLP, we de-sign a modified LP-rounding algorithm for the FLP to findan initial subset R1 ⊆ V of facilities. Without loss of gen-erality, we assume that du = 1. Then the FLP has a naturalinteger programming formulation,

min∑v∈V

f v yv +∑v∈V

∑u,v∈V

cuv xuv

s.t.∑v∈V

xuv � 1, ∀u ∈ V ,

xuv � yv , ∀u, v ∈ V ,

xuv , yv ∈ {0,1}, ∀u, v ∈ V . (3.1)

Relaxing the integrality requirement of the variables xuv

and yv , we obtain the LP relaxation of (3.1). The dual ofthe LP relaxation is

max∑u∈V

αu

s.t.∑u∈V

βuv � f v , ∀v ∈ V ,

αu − βuv � cuv , ∀u, v ∈ V ,

αu, βuv � 0, ∀u, v ∈ V . (3.2)

Let (x∗, y∗) be an optimal primal solution of theLP relaxation and (α∗, β∗) be the corresponding opti-mal dual solution. We introduce some notations withrespect to the fractional optimal solution. Denote the fa-cility cost F ∗ := ∑

v∈V f v y∗v and the connection cost

S∗ := ∑u,v∈V cuv x∗

uv . Each client u has its cost share α∗u

in the total cost, which may be divided into the fractionalconnection cost S∗

u := ∑v∈V cuv x∗

uv , and the fractional fa-cility cost F ∗

u := α∗u − S∗

u . We say that an algorithm isa (λ f , λs)-approximation algorithm if the solution it de-livers has total cost at most λ f F ∗ + λs S∗ (cf. [5]), and wecall (λ f , λs) the bifactor of the algorithm.

3.2. Algorithm

Now we are ready to present the following approxima-tion algorithm ALGFLP(γ ) for the FLP:

Algorithm 3.1. (ALGFLP(γ ))

Step 1. Constructing a complete solution

Y. Li et al. / Information Processing Letters 112 (2012) 361–364 363

Step 1.1. Solve the LP relaxation of the FLP to obtainan optimal solution (x∗, y∗).

Step 1.2. Scale up the value of the facility openingvariables y by a constant γ > 1. Then changethe value of the x-variables so as to use theclosest possible fractionally open facilities.

Step 1.3. If necessary, split facilities to obtain a com-plete solution (x, y) (i.e., there are no u, v ∈V such that 0 < xuv < yv , cf. [19] for a moredetailed argument).

Step 2. Random roundingStep 2.0. Denote the following notations: for each

client u ∈ V :• NC (u) := {v | xuv > 0}: the set of close faci-

lities of u;• N D(u) := {v | xuv = 0 and x∗

uv > 0}: the setof distant facilities of u;

• DCav(u) := ∑

v∈NC (u) cuv xuv : average distan-ce to close facilities;

• DCmax(u) := maxv∈NC (u){cuv}: maximal dis-

tance to close facilities;• D D

av(u) := ∑v∈N D (u) cuv x∗

uv : average distan-ce to distant facilities.

Step 2.1. Initially, all clients are unclustered. DenoteV := V . In the unclustered clients, chooseu := arg minu∈V DC

max(u) as the cluster centerof the current iteration. Allocate the facilitiesin NC (u) to the cluster centered by u. Re-move all clients connected to NC (u) and ufrom V . Repeat the processing until V = ∅.

Step 2.2. For every cluster center u, randomly openone of its close facilities with probabilitiesxuv . For each facility v that is not a closefacility of any cluster center, open it indepen-dently with probability yv . Define R1 as theset of open facilities. Connect each client toits closest facility in R1.

Remark 3.2. The only difference between Algorithm 3.1and Byrka and Aardal’s algorithm in [2] lies on the choiceof cluster center in Step 2.1. This new cluster centerselection strategy allows us to select a larger value ofγ (� 2.1402) to balance the total weighted facility opencost and shipping cost, contrast to the restriction γ < 2critical for their analysis in [2].

Through similar analysis to that by Byrka and Aardal[2], omitted here for simplicity (interested readers are re-ferred to the original [2] for details), we have

Theorem 3.3. For any γ � 2.1402, Algorithm 3.1 produces asolution with the bifactor (γ ,1 + 2

eγ ).

In fact, we can get another algorithm with the same bi-factor as that in Theorem 3.3 but for a slightly large rangeγ � 1.6774 when we combine the techniques of cost scal-ing and greedy argumentation by Charikar and Guha [5]with the LP-rounding technique by Byrka and Aardal [2].We give a sketch of the proof below for this alternativealgorithm.

Lemma 4.2 in [5] implies that:

Lemma 3.4. If there is a (γ f , γc)-approximation algorithm forthe FLP, then for every δ > 1, we can get a (γ f + ln(δ),1 +γc−1

δ)-approximation algorithm for the FLP using the techniques

of cost scaling and greedy argumentation, where δ is scaling fac-tor of open cost.

And from Byrka and Aardal [2], we have:

Lemma 3.5. There is a (γ0,1 + 2eγ0 )-approximation algorithm

for the FLP, where γ0 = 1.6774.

From these two lemmas, we get a (γ0 + ln(δ),1 + 2eγ0+ln(δ) )-

approximation algorithm for the FLP, where γ0 = 1.6774and δ > 0, i.e., we get a (γ ,1 + 2

eγ )-approximation algo-rithm for any γ � 1.6774. But this algorithm has a slightlyhigher time complexity than Algorithm 3.1 for a largervalue of γ (� 2.1402) due to the extra greedy argumen-tation.

4. Improved approximation algorithms for the α-RFLP

In this section, we present a polynomial time algorithm(denoted as ALGα-RFLP(γ )) for the α-RFLP.

Algorithm 4.1. (ALGα-RFLP(γ ))

Step 1. Apply the ALGFLP(γ )-approximation algorithm tothe auxiliary FLP to obtain an initial subset R1 of fa-cilities. For every r ∈ R1, denote N(r) as the set of allclients assigned to r.

Step 2. We redefine the facility cost and client demand asfollows,

f r :={

0, for r ∈ R1,

fr, for r /∈ R1;and

dr :={ ∑

u∈N(r)du, for r ∈ R1,

0, for r /∈ R1.

Apply the ALGα-CBP to the α-CBP equipped with theabove redefined facilities costs, client demands, andthe set R1 to return a new set R2.

Step 3. Output the set R1 ∪ R2 as the final set of open fa-cilities for the α-RFLP.

We remark that Step 2 of Algorithm 4.1 assigns to eachclient α backup facilities belonging to R2, one of whichwill serve this client in case its original facility in R1 fails.

To prove the approximation ratio of our algorithm, wequote Lemmas 2.5–2.6 in [6] below.

Lemma 4.2. (See [6].) Let C∗α-RFLP denote the optimal cost of the

α-RFLP, and C∗α-CBP(R1) denote the optimal cost of the α-CBP

defined at Step 2 of Algorithm 4.1. We have

C∗α-CBP(R1)� C∗

α-RFLP + S(R1).

Lemma 4.3. (See [6].) Let Cα-CBP(R1, R2) denote the cost on R2of the α-CBP defined at Step 2 of Algorithm 4.1. We have

Cα-RFLP(R1 ∪ R2) � CFLP(R1) + Cα-CBP(R1, R2).

364 Y. Li et al. / Information Processing Letters 112 (2012) 361–364

Theorem 4.4. Setting γ = γ1 ≈ 3.236, Algorithm 4.1 yields a5.236-approximation for the 1-RFLP.

Proof. It follows from Theorem 3.3 and Step 1 of Algo-rithm 4.1 with γ = 3.236 that the cost of the solution withrespect to R1 satisfies

F (R1) + 3S(R1) � 3.236F ∗ + 3

(1 + 2

e3.236

)C∗

= 3.236C∗FLP � 3.236C∗

1-RFLP. (4.1)

Consider the set of open facilities R1 ∪ R2. It follows frompart (1) of Lemma 2.1, Lemmas 4.2–4.3, and (4.1) that

C1-RFLP(R1 ∪ R2) � CFLP(R1) + C1-CBP(R1, R2)

� CFLP(R1) + 2C∗1-CBP(R1)

� CFLP(R1) + 2(C∗

1-RFLP + S(R1))

= F (R1) + 3S(R1) + 2C∗1-RFLP

� 5.236C∗1-RFLP. �

Similar to the proof of Theorem 4.4, but with part (2)of Lemma 2.1, we have an analogous approximation resultfor the case where α � 2.

Theorem 4.5. Setting γ = 6.03, Algorithm 4.1 yields a 1.005 +6.015α-approximation algorithm for the α-RFLP (α � 2).

5. Concluding remarks

We believe that our modified LP-rounding approxima-tion algorithm for the FLP developed in this work may findapplications in other facility location models. Moreover, itis also of interest to design approximation algorithm forthe multilevel and the capacitated versions of the α-RFLP,respectively.

Acknowledgements

We would like to thank the area editor and threeanonymous referees for their constructive comments thatled to this improved version. In particular, one of thereferees pointed out a technical error in an earlier ver-sion of Theorem 3.3. The research of the second authoris supported by NSF of China (No. 60773185), Beijing Nat-ural Science Foundation (No. 1102001), and Scientific Re-search Common Program of Beijing Municipal Commissionof Education (No. KM201210005033). The third author’sresearch is supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC) grant 283103.The fourth author’s research is supported by the NationalBasic Research Program of China (No. 2010CB732501).

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