Front. Math. China 2011, 6(5): 957–964DOI 10.1007/s11464-011-0153-6

A primal-dual approximation algorithmfor stochastic facility location problemwith service installation costs

Xing WANG1, Dachuan XU2, Xinyuan ZHAO2

1 Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China2 Department of Applied Mathematics, Beijing University of Technology, Beijing 100124,

China

c© Higher Education Press and Springer-Verlag Berlin Heidelberg 2011

Abstract We consider the stochastic version of the facility location problemwith service installation costs. Using the primal-dual technique, we obtain a7-approximation algorithm.

Keywords Stochastic facility location problem, primal-dual, approximationalgorithmMSC 68W25, 90C27

1 Introduction

The facility location problem (FLP) is one of the most extensively studiedcombinatorial optimization problems. The first constant approximationalgorithm for this problem appeared in [13], and the currently bestapproximation ratio is 1.488 [9]. On the negative side, one cannot obtain anapproximation algorithm with a ratio lower than 1.463 for the FLP unlessP = NP [6]. We refer to [1–5,7,8,10,14,15,20–22] for more research on theFLP along with its variants.

The facility location problem with service installation costs (FLPSC) wasfirstly introduced in [12], which gave a primal-dual 6-approximation algorithmand an LP rounding 2.391-approximation for the slightly restricted case: theinstallation costs depend only on the type of service. The approximation ratioof the later case was improved to 1.808 [18] by adopting a new way of analysisto enhance the estimation on the connection cost.

The stochastic facility location problem (SFLP) was introduced in [11] which

Received October 6, 2010; accepted July 14, 2011Corresponding author: Dachuan XU, E-mail: xudc@bjut.edu.cn

958 Xing WANG et al.

gave an LP rounding 8-approximation algorithm. The approximation ratio wasimproved to 1.86 using primal-dual method together with greedy augmentation[19]. Wang et al. [16,17] considered the k-level version and gave an LP rounding4-approximation algorithm and a primal-dual 7-approximation algorithm.

In this paper, we are interested in the stochastic version of the FLPSC,denoted as SFLPSC, which is a two stage optimization problem. The SFLPSCcan be stated formally as follows. We are given a set F of facilities, a set Dof clients, a set S of services, opening costs f0

i and fki (k ∈ {1, 2, . . . ,K}),

respectively, in the first stage and the kth scenario in the second stage for eachfacility i ∈ F, service installation costs f0s

i for service s ∈ S at facility i ∈ F inthe first stage, fks

i (k ∈ {1, 2, . . . ,K}) for service s ∈ S at facility i ∈ F in thesecond stage, metric connection cost cij between client j ∈ D and facility i ∈ F.Each client j requests a specific service g(j) ∈ S. Suppose that the occurrenceprobability of the kth scenario is pk. There is a set Dk of clients that are activein the kth scenario. We would like to open a set F0 of facilities in the first stage,and plan to open a set Fk of facilities in the second stage in the case of the kthscenario, and assign each client j ∈ Dk to an open facility i ∈ F0 or Fk whichis equipped with service g(j) ∈ S such that the total cost including facilityopening cost, service installation cost, and client connection cost is minimized.

By integrating several techniques of [8,11,12], we obtain a primal-dual7-approximation algorithm. The rest of this paper is organized as follows. InSection 2, we give the formulation for the SFLPSC. In Section 3, we present theprimal-dual algorithm. The analysis of our algorithm is offered in Section 4.Concluding remarks are given in Section 5.

2 Preliminaries

Let D := {(j, k) : j ∈ Dk, k = 1, . . . ,K}. We call (j, k) ∈ D a client-scenariopair. Let F := {(i, t) : i ∈ F, t = 0, 1, . . . ,K}. We call (i, t) ∈ F a facility-scenario pair. Denote p0 := 1. We introduce the connection cost betweenfacility-scenario pair (i, t) and client-scenario pair (j, k) as follows:

ctkij =

{cij , t = 0 or k,

+∞, otherwise.

The SFLPSC can be formulated as the following integer program:

min∑

(i,t)∈F

∑(j,k)∈D

pkctkij xtk

ij +∑

(i,t)∈F

ptfti y

ti +

∑(i,t)∈F

∑s∈S

ptftsi yts

i

s.t.∑

(i,t)∈F

xtkij � 1, ∀ (j, k) ∈ D ,

xtkij � yt

i , ∀ (i, t) ∈ F , ∀ (j, k) ∈ D ,

xtkij � y

tg(j,k)i , ∀ (i, t) ∈ F , ∀ (j, k) ∈ D ,

xtkij , yt

i , ytsi ∈ {0, 1}, ∀ (i, t) ∈ F , ∀ (j, k) ∈ D , s ∈ S.

(1)

Approximation algorithm for stochastic facility location problem 959

We remark that g(j, k) = g(j), ∀ (j, k) ∈ D . In the above integer program,the binary variable yt

i indicates whether facility-scenario pair (i, t) ∈ F isopened, yts

i indicates whether service type s is installed at facility-scenariopair (i, t), xtk

ij indicates whether client-scenario pair (j, k) ∈ D is connected tofacility-scenario pair (i, t) ∈ F . The linear programming relaxation of (1) is

min∑

(i,t)∈F

∑(j,k)∈D

pkctkij xtk

ij +∑

(i,t)∈F

ptfti y

ti +

∑(i,t)∈F

∑s∈S

ptftsi yts

i

s.t.∑

(i,t)∈F

xtkij � 1, ∀ (j, k) ∈ D ,

xtkij � yt

i , ∀ (i, t) ∈ F , ∀ (j, k) ∈ D ,

xtkij � y

tg(j,k)i , ∀ (i, t) ∈ F , ∀ (j, k) ∈ D ,

xtkij , yt

i , ytsi � 0, ∀ (i, t) ∈ F , ∀ (j, k) ∈ D , s ∈ S.

(2)

Let Gs be the set of client-scenario pairs requesting service s. The dual of (2)can be written as follows:

max∑

(j,k)∈D

αkj

s.t. αkj − βtk

ij − θtkij � pkc

tkij , ∀ (i, t) ∈ F , ∀ (j, k) ∈ D ,∑

(j,k)∈D

βtkij � ptf

ti , ∀ (i, t) ∈ F ,

∑(j,k)∈Gs

θtkij � ptf

tsi , ∀ (i, t) ∈ F , ∀ s ∈ S,

αkj , β

tkij , θtk

ij � 0, ∀ (i, t) ∈ F , ∀ (j, k) ∈ D .

(3)

3 Algorithm

In this section, we present the primal-dual algorithm for the SFLPSC. For anygiven t = 0, 1, . . . ,K, let us denote Ft := {(i, t) | i ∈ F}. Since the generalSFLPSC is as hard as the set-cover problem, we consider the restricted versionunder the following assumption ([12]).

Assumption 1 For any t = 0, 1, . . . ,K, there is an ordering (denoted as Ot)on the facility-scenario pairs in Ft such that if (i, t) comes before (i′, t) in thisordering, then f ts

i � f tsi′ , ∀ s ∈ S.

Following the approaches of [8,12,17], we present the primal-dual algorithmwhich is associated with a so-called time variable τ as follows.

Algorithm 1

Step 0 Initialization We start at time τ := 0. Set D := D , F := ∅,F

s := ∅, F t := ∅, and Fst := ∅ (t = 0, 1, . . . ,K, s ∈ S). For every (j, k) ∈ D ,

(i, t) ∈ F , initialize αkj := 0, βtk

ij := 0, θtkij := 0, D(i, t) := ∅, and S(i, t) := ∅.

960 Xing WANG et al.

Step 1 Constructing a dual solution

Step 1.1 At time τ, set αkj := pkτ for every client-scenario pair (j, k) ∈ D . We

increase the time τ until one of the following events happens.

Event 1. There exist (j, k) ∈ D and (i, t) ∈ F such that αkj = pkc

tkij .

Set D(i, t) := D(i, t) ∪ {(j, k)}. If g(j, k) /∈ S(i, t), then start increasingθtkij as the same rate as αk

j , and set θtkij := pk(τ − ctk

ij ). If g(j, k) ∈ S(i, t) and(i, t) /∈ F t, then instead increase βtk

ij as the same rate as αkj , set βtk

ij := pk(τ−ctkij ).

If g(j, k) ∈ S(i, t) and (i, t) ∈ F t, set D := D \ {(j, k)}.Event 2. There exist (i, t) ∈ F and s ∈ S such that

∑(j,k)∈Gs

θtkij = ptf

tsi .

Fix τ tsi := τ and set S(i, t) := S(i, t)∪{s}. If (i, t) ∈ F t, set D := D \ (Gs ∩

D(i, t)). If (i, t) /∈ F t, fix θtkij and then instead increase βtk

ij as the same rate asαk

j , set βtkij := pk(τ − τ ts

i ) for each (j, k) ∈ D ∩ Gs ∩ D(i, t).

Event 3. There exists (i, t) ∈ F such that∑

(j,k)∈D βtkij = ptf

ti .

Fix τ ti := τ and set F t := F t ∪ {(i, t)}. Fix βtk

ij for each (j, k) ∈ D . Set

D := D \ {(j, k) ∈ D | (j, k) ∈ D(i, t), g(j, k) ∈ S(i, t)}.Step 1.2 If D �= ∅, go to Step 1.1. Otherwise, set

F := F 0 ∪ F 1 ∪ · · · ∪ F K ,

Fst := {(i, t) | (i, t) ∈ F t, s ∈ S(i, t)} (t = 0, 1, . . . ,K, s ∈ S),

Fs := F

s0 ∪ F

s1 ∪ · · · ∪ F

sK (s ∈ S).

Step 2 Constructing a primal integer solution

Step 2.1 Opening facility-scenario pairs. For each t = 0, 1, . . . ,K, let F′t := ∅.

Consider the facility-scenario pair (i, t) ∈ F t in the order of Ot. If βtkij = 0 for

each (j, k) ∈ {(j, k) | βtki′j > 0, (i′, t) ∈ F

′t}, set F

′t := F

′t ∪ {(i, t)}. We open the

facility-scenario pairs in F′t (t = 0, 1, . . . ,K).

Step 2.2 Installing servicesStep 2.2.1 For each t = 0, 1, . . . ,K and s ∈ S, let F s

t := ∅. Consider thefacility-scenario pair (i, t) ∈ F

st in a particular order: facility-scenario pairs in

Fst ∩F

′t in increasing order of τ ts

i , and then facility-scenario pairs in Fst −F

′t in

increasing order of τ ti . If θtk

ij = 0 for each (j, k) ∈ {(j, k) | θtki′j > 0, (i′, t) ∈ F s

t },set F s

t := F st ∪ {(i, t)}.

Step 2.2.2 For all facility-scenario pairs in F st ∩F

′t, we install service s at them.

For each (i, t) ∈ F st − F

′t, we pick a facility-scenario pair (i′, t) ∈ F

′t such that

βtkij , βtk

i′j > 0 for some (j, k) and (i′, t) � (i, t) (in the ordering of Ot), and installservice s on facility-scenario pair (i′, t).Step 2.3 Assigning demands. Assign each (j, k) ∈ D to the nearest (withrespect to connection cost) open facility-scenario pair which equipped withservice g(j, k).

Approximation algorithm for stochastic facility location problem 961

4 Analysis

Let us denote

F′ := F

′0 ∪ F

′1 ∪ · · · ∪ F

′K ,

D ′ := {(j, k) ∈ D : ∃ (i, t) ∈ F′ such that βtk

ij > 0}.The analysis of Algorithm 1 is proceeded as follows. First, we bound the

facility-scenario pair opening cost. Second, we bound the service installationcost. Third, we estimate the client-scenario pair connection cost in two caseswith respect to D ′. Finally, we add up all the above costs and draw our mainresult.

Lemma 1 The cost of opening facility-scenario pairs is at most 2∑

(j,k)∈D ′ αkj .

Proof It follows from Step 2.1 of Algorithm 1 that the set of opening facility-scenario pairs is F

′. From the definition of ctk

ij , we have βtkij = 0, ∀ t �= 0, k.

Thus, we have ∑(i,t)∈F

′ptf

ti =

∑(i,t)∈F

′

∑(j,k)∈D ′

βtkij

=∑

(j,k)∈D ′

∑(i,t)∈F

′βtk

ij

=∑

(j,k)∈D ′

( ∑(i,0)∈F

′0

β0kij +

∑(i,k)∈F

′k

βkkij

)

� 2∑

(j,k)∈D ′αk

j . �

Lemma 2 The cost of installing services is at most 2∑

(j,k)∈D αkj .

Proof From the definition of ctkij , Step 2.2 of Algorithm 1, and Assumption 1,

we have the cost of installing services is no more than

∑s∈S

K∑t=0

∑(i,t)∈F s

t

ptftsi =

∑s∈S

K∑t=0

∑(i,t)∈F s

t

∑(j,k)∈Gs

θtkij

=∑s∈S

∑(j,k)∈Gs

K∑t=0

∑(i,t)∈F s

t

θtkij

=∑

(j,k)∈D

( ∑(i,0)∈F

g(j)0

θ0kij +

∑(i,k)∈F

g(j)k

θkkij

)

� 2∑

(j,k)∈D

αkj . �

962 Xing WANG et al.

Lemma 3 If (j, k) ∈ D ′, then the client connection cost incurred for (j, k) isat most 3αk

j .

Proof Consider (j, k) ∈ D ′ with g(j, k) = s. Recall the definition of D ′. Let(i, t) ∈ F

′t (t = 0 or k) be the facility-scenario pair that (j, k) is connected to,

and we have βtkij > 0. We consider the following two cases.

Case 1 (i, t) ∈ F st .

This means that we have installed service s at (i, t), therefore, (j, k) connectsto a facility-scenario pair which is open and installed service g(j, k). Thus,

pkctkij � αk

j − βtkij � αk

j .

Case 2 (i, t) /∈ F st .

It follows from Step 2.2.1 of Algorithm 1 that there exist (i′, t) ∈ F′t ∩ F s

t

and (j′, k′) ∈ Gs such that τ tsi′ � τ ts

i , θtk′ij′ , θ

tk′i′j′ > 0, which imply ctk′

ij′ � τ tsi

and ctk′i′j′ � τ ts

i′ . We connect (j, k) to (i′, t) which is open and installed services = g(j, k). Combined with the triangle inequality and βtk

ij > 0, we have

pkctki′j � pk(ctk′

i′j′ + ctk′ij′ + ctk

ij )

� pk(τ tsi′ + τ ts

i ) + pkctkij

� 2pkτtsi + pkc

tkij

� 3αkj .

Summing up the above two cases, we conclude the proof. �Lemma 4 If (j, k) /∈ D ′, then the client connection cost incurred for (j, k) isat most 5αk

j .

Proof Consider a client-scenario pair (j, k) /∈ D ′, and let s = g(j, k). Let (i, t)(t = 0 or k) be the facility-scenario pair that caused (j, k) to be excluded fromD ′. Then we have αk

j = max{pkctkij , pkτ

ti , pkτ

tsi }. We consider the following cases.

Case 1 (i, t) ∈ F′t ∩ F s

t .Connect (j, k) to (i, t) which is open and installed service s = g(j, k).

Obviously, we havepkc

tkij � αk

j .

Case 2 (i, t) ∈ F st − F

′t.

It follows from Step 2.2.1 of Algorithm 1 that there exist (i′, t) ∈ F′t (which

is installed service s) and (j′, k′) ∈ D such that βtk′ij′ , β

tk′i′j′ > 0. We connect

(j, k) to (i′, t) which is open and installed service s = g(j, k). By the triangleinequality, we have

pkctki′j � pk(ctk

ij + ctk′ij′ + ctk′

i′j′) � pkctkij + 2pk min{τ t

i′ , τti } � pkc

tkij + 2pkτ

ti � 3αk

j .

Case 3 (i, t) /∈ F st .

Approximation algorithm for stochastic facility location problem 963

It follows from (i, t) ∈ Fst and Step 2.2.1 of Algorithm 1 that there exist

(i′, t) ∈ F st and (j′, k′) ∈ D such that θtk′

ij′ , θtk′i′j′ > 0. From Step 1 of Algorithm

1, we obtain

max{ctk′ij′ , c

tk′i′j′} �

αk′j′

pk′� max{τ t

i , τtsi } �

αkj

pk. (4)

Case 3.1 (i′, t) ∈ F′t. From Step 2.2.2 of Algorithm 1, (i′, t) is installed service

s. We connect (j, k) to (i′, t). Combining the triangle inequality and (4), we get

pkctki′j � pk(ctk

ij + ctk′ij′ + ctk′

i′j′) � 3αkj .

Case 3.2 (i′, t) /∈ F′t. From Step 2.2.1 of Algorithm 1, we have τ t

i′ � τ ti . It

follows from Step 2.2.2 of Algorithm 1 that there exist (i′′, t) ∈ F′t (which

installed service s) and (j′′, k′′) ∈ D such that βtk′′i′j′′ , β

tk′′i′′j′′ > 0. We connect

(j, k) to (i′′, t). Combining the triangle inequality and (4), we get

pkctki′′j � pk(ctk

ij + ctk′ij′ + ctk′

i′j′ + ctk′′i′j′′ + ctk′′

i′′j′′)

� pkctkij + pk(ctk′

ij′ + ctk′i′j′) + 2pk min{τ t

i′ , τti′′}

� αkj + 2αk

j + 2pkτti

� 5αkj .

Summing up the above three cases, we conclude the proof. �Theorem 1 Algorithm 1 yields a 7-approximation algorithm for the SFLPSC.

Proof It follows from Lemmas 1–4 that the total cost including opening cost,installing service cost, and connecting cost is bounded by

2∑

(j,k)∈D ′αk

j + 2∑

(j,k)∈D

αkj + 3

∑(j,k)∈D ′

αkj + 5

∑(j,k)∈D\D ′

αkj = 7

∑(j,k)∈D

αkj . �

5 Discussion

If the installation costs depend only on the type of service, there is an LProunding 1.808-approximation algorithm for the FLPSC [18]. It will beinteresting to further improve the approximation ratio for the SFLPSC.

Acknowledgements The authors would like to thank two anonymous referees for their

helpful comments. This work was partially done while the first author was a visiting doctorate

student at the Department of Applied Mathematics, Beijing University of Technology and

supported in part by the National Natural Science Foundation of China (Grant No. 60773185).

The research of the second author was supported by the National Natural Science Foundation

964 Xing WANG et al.

of China (Grant Nos. 60773185, 11071268), the Natural Science Foundation of Beijing (No.

1102001), and PHR (IHLB).

References

1. Ageev A, Ye Y, Zhang J. Improved combinatorial approximation algorithms for thek-level facility location problem. SIAM J Discrete Math, 2004, 18: 207–217

2. Byrka J, Aardal K I. An optimal bifactor approximation algorithm for the metricuncapacitated facility location problem. SIAM J Comput, 2010, 39: 2212–2213

3. Chen X, Chen B. Approximation algorithms for soft-capacitated facility location incapacitated network design. Algorithmica, 2009, 53: 263–297

4. Du D, Lu R, Xu D. A primal-dual approximation algorithm for the facility locationproblem with submodular penalties. Algorithmica, DOI: 10.1007/s00453-011-9526-1

5. Du D, Wang X, Xu D. An approximation algorithm for the k-level capacitated facilitylocation problem. J Comb Optim, 2010, 20: 361–368

6. Guha S, Khuller S. Greedy strike back: improved facility location algorithms.J Algorithms, 1999, 31: 228–248

7. Jain K, Mahdian M, Markakis E, Saberi A, Vazirani V V. Greedy facility locationalgorithms analyzed using dual fitting with factor-revealing LP. J ACM, 2003, 50:795–824

8. Jain K, Vazirani V V. Approximation algorithms for metric facility location andk-median problems using the primal-dual schema and Lagrangian relaxation. J ACM,2001, 48: 274–296

9. Li S. A 1.488-approximation algorithm for the uncapacitated facility location problem.In: Proceedings of ICALP. 2011, Part II: 77–88

10. Mahdian M, Ye Y, Zhang J. Approximation algorithms for metric facility locationproblems. SIAM J Comput, 2006, 36: 411–432

11. Ravi R, Sinha A. Hedging uncertainty: approximation algorithms for stochasticoptimization programs. Math Program, 2006, 108: 97–114

12. Shmoys D B, Swamy C, Levi R. Facility location with service installation costs. In:Proceedings of SODA. 2004, 1081–1090

13. Shmoys D B, Tardos E, Aardal K I. Approximation algorithms for facility locationproblems (extended abstract). In: Proceedings of STOC. 1997, 265–274

14. Shu J. An efficient greedy heuristic for warehouse-retailer network design optimization.Transport Sci, 2010, 44: 183–192

15. Shu J, Teo C P, Shen Z J Max. Stochastic transportation-inventory network designproblem. Oper Res, 2005, 53: 48–60

16. Wang Z, Du D, Gabor A F, Xu D. An approximation algorithm for solving the k-levelstochastic facility location problem. Oper Res Lett, 2010, 38: 386–389

17. Wang Z, Du D, Xu D. A primal-dual approximation algorithm for the k-level stochasticfacility location problem. In: Proceedings of AAIM. 2010, 253–260

18. Xu D, Zhang S. Approximation algorithm for facility location problems with serviceinstallation costs. Oper Res Lett, 2008, 36: 46–50

19. Ye Y, Zhang J. An approximation algorithm for the dynamic facility location problem.In: Combinatorial Optimization in Communication Networks. Dordrecht: KluwerAcademic Publishers, 2005, 623–637

20. Zhang J. Approximating the two-level facility location problem via a quasi-greedyapproach. Math Program, 2006, 108: 159–176

21. Zhang J, Chen B, Ye Y. A multiexchange local search algorithm for the capacitatedfacility location problem. Math Oper Res, 2005, 30: 389–403

22. Zhang P. A new approximation algorithm for the k-facility location problem. TheorComput Sci, 2007, 384: 126–135