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A 5-approximation for facility location with

non-uniform capacities

1, Manisha Bansal2, Naveen Garg1, and Neelima Gupta2

1 Indian Institute of Technology Delhi2 University of Delhi

Abstract. In this paper, we propose and analyse a local search algo-rithm for approximating non-uniform capacitated facility location prob-lem which is a modification of the algorithm for the same problem byZhang et al [7]. They showed that the solution obtained by their al-gorithm is within a factor of (6,5) of the value of an optimal solution.

We improve this approximation ratio to (5,5). We modify their closeand multi operations such that apart from the demand of facilities be-ing closed, some more demand served by other facilities in the currentsolution can be reassigned to utilize the facilities opened by the opera-tion in a better/efficient way. The idea of taking linear combinations ofinequalities used in Aggarwal et al [1] is crucial in achieving this result.The analysis is tight for this algorithm.

1 Introduction

In a facility location problem we are given a set of clients C and facility locationsF. Each facility i F has a facility cost fi (the facility cost). The cost of servicinga client j by a facility i is given by ci,j (the service cost) and these costs form ametric i.e. for facilities i, i and clients j, j, ci,j ci,j+ci,j+ci,j. The objectiveis to select a subset S F , so that the total cost for opening the facilities andfor serving all the clients is minimized. A client can be served by an open facilityonly.

When the number of clients that a facility i can serve is bounded by ui, wehave capacitated facility location problem(CFLP). Ifui is same for all the facilitiesthen the problem is called Uniform capacitated facility location problem. We areinterested in Non-uniform capacitated facility location problem, in which capacitymay be different for different facilities. We do not require that all the demandfrom a client be served by a single facility.

For the problem ofuniform capacities Korupolu et al [3] showed that a simplelocal search heuristic proposed by Kuehn and Hamburger [4] provides a constant

factor approximation. Chudak and Williomson [2] improved on the analysis ofthe same heuristic to show that this local search algorithm gives a 5 .83 ap-proximation. Aggarwal et al [1] further improved the factor to 3 in which they

Work done as part of the Approximation Algorithms partner group of MPI-Informatik, Germany

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introduced a new approach for analysis in which they take a linear combinationof ineqalities( capturing local optimality) instead of just taking sum of theseineqalities.

We have used similar ideas for the problem of non-uniform capacities. Wealso modify close and multi operations of Zhang et al such that apart from thedemand of facilities being closed, some more demand served by other facilitiesin the current solution can be reassigned to utilize the facilities opened by theoperation in a better/efficient way. Whereas in all prvious work on CFLP withnon uniform capacities, reassignment of clients is done for only those clientswhich were being served by the facilities closed in an operation. Also whileanalysing the cost of an operation we assign clients fractionally to the facilities.This assignment cannot be better than the optimum assignment. The first localseacrch heuristic for non-uniform capacities problem was due to Pal, Tardos andWexler [6] which yeilds an 8.53 factor approximation algorithm for this problem.Mahdian and Pal [5] reduced this upper bound to 7.88 which was further reduced

by Zhang et al [7] to 5.83.In this paper, we present a 5-approximation algorithm for this problem. Theremainder of this paper is organised as follows. In section 2 we briefly describethe algorithm and analysis of Zhang et al. In section 3 we bound the facility costof our solution. We also present a tight example in section 4.

2 Preliminaries

For a given subset S F, the optimal assignment of clients to the set of facilitiesin S can be computed by solving a mincost flow problem. Therefore to solveCFLP with non-uniform capacities we only require to determine a good subsetS F of facilities. Let S also denote the solution consisting of facility set S. We

denote the cost of solution S by C(S).Pal et al suggested a local search algorithm to find a good approximate

solution for CFLP with non-uniform capacities. Following are the operations intheir algorithm:

Add(s): For s F, S S+ {s} and reassign all clients to S by solvingassignment problem. The cost of this operation is C(S) C(S).

open(s,T): Open one facility s F and close a set of facilities T S {s}and reassign all the clients served by facilities in T to s. Clients served byfacilities in S \ T are not affected. Cost of reassigning each client j whichwas being served by t T to s is atmost cts. Therefore cost of the operationis f(s)

tT f(t) + , is the total cost of reassigning all the clients of

closed facilities.

close(s,T): Close one facility s, s S. Open a set of facilities T F {s}.Assign all the clients from s to facilities in T. Clients served by facilities inS\{s} are not affected. Cost of reassigning a client j which was being servedby s to t T is atmost cst. Therefore cost of the operation is

tT f(t)

f(s) + , is the total cost of reassigning all the clients of closed facility s.

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Zhang et al added the following operation to the above set of operations:

multi(r,R,t,T): 1. Close a set of facilities R {r} S where r / R and

R may be empty. 2. Open a set of facilities ({t} T) (F R {r}) 3.Assign all the demand from r to T {t} and from R to t . As describedin open and close operations, the cost of reassigning a client j served by rto a facility t T {t} is atmost crt and the cost of reassigning a clientfrom a facility s R to t is atmost cst. Therefore cost of the operation is

t{t}T f(t)

sR{r} f(s) + , is the total cost of reassigning allthe clients of closed facilities to the opened facilities.

S is locally optimum if none of the four operations are possible and at thispoint the algorithm stops. Polynomial running time can be ensured at the ex-pense of an additive in the approximation factor by doing a local search oper-ation only if the cost reduces by more than an 1 /n factor, for > 0.

Let S F be a local optimal solution and O F be a global optimal

solution. Cost of a solution A F, C(A) = Cf(A) + Cs(A), where Cf(A) is thefacility cost of solution A and Cs(A) is the service cost of solution A. The boundon the service cost is well established now as in Korupolu et al [3] and Mahdianet al [5]. Following lemma from Mahdian et al states the bound on service cost.

Lemma 1. The service cost Cs(S) of a local optimal solution S is bounded bythe optimal total costC(O), where O is a global optimal solution.

Pal, Tardos and Wexler [6] bound the cost of minimum cost flow in theexchange graph and also show the existence of an optimal flow which is acyclic.We reproduce their result without proof in the following lemmas.

Lemma 2. There is a flow in the exchange graph with cost atmostcs(O)+cs(S).

Lemma 3. There is a minimum cost flowy in the exchange graph whose nonzeroedges form a forest.

To bound the facility cost of S, Zhang et al used the optimal solution of atransshipment graph (as is also used by Pal et al)to identify the set of operationswhich provide desirable inequalities. Zhang et al in their analysis described a setofclose and multioperations to get an upper bound on facility cost of their localoptimal solution. We next briefly sketch the analysis of Zhang et al.

Consider a minimum cost flow y in exchange graph whose non zero edgesform a forest say G. Reorient each tree in this forest so that some vertex r Ois the root. For a vertex t, denote children of t by K(t). A vertex t O is eitherweak, if

sK(t) y(s, t) > 1/2y(., t) or strong otherwise.

Consider a tree r rooted at some vertex r O. Consider a subtree of atmost

two levels rooted at t O as in figure 1. A vertex s K(t) is heavy if y(s, t) 1/2y(., t) or light otherwise. A light facility is nondominant, if

tK(s) y(s, t) >

1/2y(s, .) and dominant otherwise. Let KND (t) be light-nondominant childrenoft and KD(t) be light-dominantchildren oft. Similarly Kw(s) denotes the weakchildren of s S and Kst(s) denotes strong children of s.

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Fig. 1. Classification of facilities

To handle light non-dominant facilities KND (t) they arrange these facilitiesaccording to the nondecreasing order of rem(si) values, where

rem(si) = max{0, y(si, t)

tKw(si)y(si, t

)}

Except for sk for all other si KND (t) Zhang et al perform close(si, T)operation where T = K(si) Kst(si+1). In this operation strong children of siand si+1 i.e. t

Kst(si) Kst(si+1) get atmost y(si, t) clients from si. Weakchildren ofsi i.e t

Kw(si) get atmost 2y(si, t) clients in this operation. Figure2 illustrates this operation.

For the remaining facilities in K(t) Zhang et al consider three cases dependingupon whether t is weak or strong.

Case 1: t is strong. t being strong doesnt have a heavy child. Zhang et alclose facilities in KD(t)sk in a multi opetation multi(sk, KD(t), t , T) whereT = K(sk), refer figure 3. A facility in s KD(t) sends all its clients to tthus using the edge (s, t) to send atmost 2y(s, t) clients in the operation.Facility sk sends y(sk, t

) clients to each t T {t}.Case2: t is weak, and has a heavy child t has exactly one heavy child say

sh, refer figure ??. They perform close(sh, T) where T = k(sh) {t} to closeheavy facility and send y(sh, t

) clients from sh to t T. For the remainingfacilities in K(t) they close facilities in KD(t) sk in a multi opetation

multi(sk, KD(t), t , T ) where T = K(sk). A facility in s KD(t) sends all itsclients to t thus using the edge (s, t) to send atmost 2y(s, t) clients in theoperation. Facility sk sends y(sk, t

) clients to each t T {t}.Case 3: t is weak and doesnt have a heavy child For this case they

partition facilities in KD(t) {sk} into two subsets such that the facilities

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Fig. 2. Illustration ofclose(si,Kst(si) Kst(si+1))

Fig. 3. Illustration ofmulti(sk,KD(t), t ,K(sk))

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Fig. 4. Illustration of case 2

in each of these subsets can be closed in a multioperation, thus needing twomulti operations to handle facilities in KD(t) {sk}. Figure ?? depicts howthe facilities are partitioned.For facilities s Di {i},i = 1, 2, they perform multi(i, Di, t , T i) whereTi = k(i).

By defining the operations in above manner for different types of facilities ofS, Zhang et al achieve the following:

Lemma 4. (i) Each facilitys S is closed exactly once. (ii) Each facility o Ois opened atmost thrice. (iii) The total cost of reassignment of all the operationsis bounded by 2

sSO

tO c(s, t)y(s, t).

We observe that in all the operations described above, a facility of t Owhich is opened in an operation is not being utilized to its full capacity, butevery time it is opened it pays facility cost in full. Infact the total extent uptowhich a facility t O is going waste is atleast equal to its capacity in manycases. We have used this observation to modify their close operation such thatthis modified close operation mclose(s,T) uses each facility opened to maximumpossible extent thus not wasting any capacity. Thus whenever a facility of O isopened it pays full facility cost and is also utilized upto the full capacity.

3

In section 2 we briefly discussed the analysis of Zhang et al and also observed thatthe operations that they used for their analysis did not utilize full capacity of afacility of optimum when it was opened in an operation. We modified their close

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Fig. 5. Illustration of case 3

operation and called it mclose which was able to utilize every facility openedin an operation to the full capacity. We also modified their multi operationwhich generalized their open and close operations by assuming it now to be ageneralization of open and mclose operations, and this way multi operation isalso able to utilize every facility opened in the new multi operation to the fullcapacity.

Let R O be the set of facilities which were considered exactly thrice overall the inequalities and O R be the set of facilities considered atmost twice.

Lemma 5. The total number of clients of NO(t)t R which are assigned to tis greater than or equal to ut.

Proof. Case 1: If t is strong: t participates in close/multi operations w.r.t.three sets of facilities of S.

1. t participates in close operations w.r.t. closing of facilities in set {si}.2. t participates in close operations w.r.t. closing of facilities in set {si1}.3. t participates in close operations w.r.t. closing of facilities in set KD(t)

{sk}.In the previous section we showed that if Es,t is the amont of flow that treceived from s then t also receives ut Es,t amount of clients from NO(t).

1. t receives Esi,t = y(si, t) from si2. t receives Esi1,t y(si, t) from si13. t receives

s

{sk

}KD(t)

Es,t = y(sk, t)+2

s

KD(t)

y(s, t)from {sk}

KD(t)Therefore total clients t receives from NO(t) in three operations is atleast

ut y(si, t) + ut y(si, t) + ut y(sk, t) 2

sKD(t)y(s, t)

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= 3ut 2y(si, t) y(sk, t) 2

sKD(t)y(s, t) (1)

Also note that

y(si, t) + y(sk, t) +

sKD(t)y(s, t) y(., t) ut .

Fig. 6. Operations for strong facility

therefore 1 can be written as

3ut 2ut = ut

If t is weak:If t has a heavy child say sh = 2 then D2 is empty.As discussed in case when t is strong, in this case also t participates in

close/multi operations w.r.t. three sets of facilities of S.1. t participates in close operations w.r.t. closing of facilities in set {s}.2. t participates in close operations w.r.t. closing of facilities in set D1{1}.3. t participates in close operations w.r.t. closing of facilities in set D2{2}.

Next we specify Es,t values for edges adjacen to t in G.

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Fig. 7. Operations for weak facility

1. t receives Es,t 2y(s, t) from s.2. t receives

s{1}D1 Es,t y(1, t) + 2

sD1 y(s

, t)from D1 {1}.3. t receives

s{2}D2 Es,t y(2, t) + 2

sD2 y(s

, t)from D2 {2}Therefore total clients t receives from NO(t) in three operations is atleast

ut 2y(s, t) + ut y(1, t) 2

sD1y(s, t) + ut y(2, t) 2

sD2y(s, t)

= 3ut 2y(s, t) y(1, t) y(2, t) 2

sD1y(s, t) 2

sD2y(s, t) (2)

Also note that

y(s, t) + y(1, t) + y(2, t) +

sD1D2y(s, t) y(., t) .

Therefore 2 can be written as

3ut 2ut = ut. (3)

4 Tight example

Zhang et al in [7] claimed that their analysis is tight by giving a tight example.We first describe that example given in figure 8 and then show that their claim

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is not true. The example given in figure has 2n square facilities each with facilitycost 4 and capacity n, 2n circle facilties each with facility cost 0 and capacityn 1 and one circle facility with facility cost 4 and capacity 2 n. There are total2n clients with n 1 demand each and 2n clients with 1 demand each. Numberson the edges represent per unit service costs. Current local optimal solution isrepresented by the set of all the square facilities and the optimal solution isrepresented by the set of all the circle facilities. Cost of the current solution is8n + 2n = 10n and cost of the optimal solution is 4 + 2n. Ratio of the costs ofthese two solutions turns out to be 5 as n goes to .

If we decrease the per unit costs on edges even slightly, then the set ofsquare facilities no longer is a local optimal solution. Note that closing onesquare facility and opening two circle facilities, where one circle facility is exactlybelow the square facility closed and another circle facility next to this circlefacility(before or after), will improve the solution consisting of all the squarefacilities by bringing reduction in the cost of the current solution. Zhang et al

claim that the per unit service costs in the figure are scaled up by a factor = 1+

2

2 and they compare the cost of the two solutions w.r.t. prescaled servicecosts. However with prescaled service costs the current solution is not localoptimal. Thus with this example they can only claim the lower bound of theiralgorithm to be 5. Hence their claim is not true.

The set of square facilities forms a local optimal solution w.r.t. mclose oper-ation.

References

1. Ankit Aggarwal, L. Anand, Manisha Bansal, Naveen Garg, Neelima Gupta, Shub-ham Gupta, and Surabhi Jain. A 3-approximation for facility location with uniformcapacities. In IPCO, pages 149162, 2010.

2. Fabian Chudak and David P. Williamson. Improved approximation algorithms forcapacitated facility location problems. Math. Program., 102(2):207222, 2005.

3. Madhukar R. Korupolu, C. Greg Plaxton, and Ra jmohan Rajaraman. Analysis ofa local search heuristic for facility location problems. J. Algorithms, 37(1):146188,2000.

4. A. A. Kuehn and M. J. Hamburger. A heuristic program for locating warehouses.Management Science, 9(9):643666, July 1963.

5. Mohammad Mahdian and Martin Pal. Universal facility location. In ESA, pages409421, 2003.

6. M. Pal, E. Tardos, and T. Wexler. Facility location with nonuniform hard capacities.In FOCS 01: Proceedings of the 42nd IEEE symposium on Foundations of ComputerScience, page 329, Washington, DC, USA, 2001. IEEE Computer Society.

7. Jiawei Zhang, Bo Chen, and Yinyu Ye. A multiexchange local search algorithm forthe capacitated facility location problem. Math. Oper. Res., 30(2):389403, 2005.

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Fig. 8. Tight example

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