LP-based Approximation Algorithms for Capacitated Facility Location

Transcription

1 LP-based Approxmaton Algorthms for Capactated Faclty Locaton Retsef Lev 1, Davd B. Shmoys 2,andChatanyaSwamy 3 1 School of ORIE, Cornell Unversty, Ithaca, NY rl227@cornell.edu 2 School of ORIE and Department of Computer Scence, Cornell Unversty Ithaca, NY shmoys@cs.cornell.edu 3 Department of Computer Scence, Cornell Unversty, Ithaca, NY swamy@cs.cornell.edu 1 Introducton There has been a great deal of recent work on approxmaton algorthms for faclty locaton problems [9]. We consder the capactated faclty locaton problem wth hard capactes. We are gven a set of facltes, F, and a set of clents D n a common metrc space. Each faclty has a faclty openng cost f and capacty u that specfes the maxmum number of clents that may be assgned to ths faclty. We want to open some facltes from the set F and assgn each clent to an open faclty so that at most u clents are assgned to any open faclty. The cost of assgnng clent j to faclty s gven by ther dstance c j, and our goal s to mnmze the sum of the faclty openng costs and the clent assgnment costs. The recent work on faclty locaton problems has come n two varetes: LPbased algorthms, and local search-based algorthms. For the problem descrbed above, no constant approxmaton algorthm based on LP s known, and n fact, no LP relaxaton s known for whch the rato between the optmal nteger and fractonal values has been bounded by a constant. Surprsngly, constant performance guarantees can stll be proved based on local search, but these have the dsadvantage that on an nstance by nstance bass, one never knows anythng stronger than the guarantee tself (wthout resortng to solvng an LP relaxaton anyway). We present an algorthm that rounds the optmal fractonal soluton to a natural LP relaxaton by usng ths soluton to gude the decomposton of the nput nto a collecton of sngle-demand-node capactated faclty locaton problems, whch are then solved ndependently. In the specal case that all faclty openng costs are equal, we show that our algorthm s a 5-approxmaton algorthm, thereby also provdng the frst constant upper bound on the ntegralty gap of ths formulaton n ths mportant specal case. One salent feature of our algorthm s that t reles on a decomposton of the nput nto nstances of the sngle-demand capactated faclty locaton problem; n ths way, the algorthm Research suported partally by a grant from Motorola and NSF grant CCR Research supported partally by NSF grant CCR

2 mrrors the work of Aardal [1], who presents a computatonal polyhedral approach for ths problem whch uses the same core problem n the dentfcaton of cuttng planes. There are several varants of the capactated faclty locaton problem, whch have rather dfferent propertes, especally n terms of the approxmaton algorthms that are currently known. One dstncton s between soft and hard capactes: n the latter problem, each faclty s ether opened at some locaton or not, whereas n the former, one may specfy any nteger number of facltes to be opened at that locaton. Soft capactes make the problem easer; Shmoys, Tardos, & Aardal [11] gave the frst constant approxmaton algorthm for ths problem based on an LP-roundng technque; Jan & Vazran [4] gave a general technque for convertng approxmaton algorthm results for the uncapactated problem nto algorthms that can handle soft capactes. Korupolu, Plaxton, & Rajaraman [5] gave the frst constant approxmaton algorthm that handles hard capactes, based on a local search procedure, but ther approach worked only f all capactes are equal. Chudak & Wllamson [3] mproved ths performance guarantee to 5.83 for the same unform capacty case. Pál, Tardos, & Wexler [8] gave the frst constant performance guarantee for the case of non-unform hard capactes. Ths was recently mproved by Mahdan & Pál [6] and Zhang, Chen, & Ye [13] to yeld a 5.83-approxmaton algorthm. There s also a dstncton between the case of unsplttable assgnments and splttable ones. That s, suppose that each clent j has a certan demand d j to be assgned to open facltes so that the total demand assgned to each faclty s at most ts capacty: does each clent need to have all of ts demand served by a unque faclty? In the former case, the answer s yes, whereas n the latter, the answer s no. All approxmaton algorthms for hard capactes have focused on the splttable case. Note that once one has decded whch facltes to open, the optmal splttable assgnment can be computed by solvng a transportaton problem. A splttable assgnment can be converted to an unsplttable one at the cost of ncreasng the requred capacty at each faclty (usng an approxmaton algorthm for the generalzed assgnment problem [10]). Of course, f there are nteger capactes and all demands are 1, there s no dstncton between the two problems. For hard capactes, t s easy to show that the natural LP formulatons do not have any constant ntegralty rato; the smplest such example has two faclty locatons, one essentally free, and one very expensve. In contrast, we focus on the case n whch all faclty openng costs are equal. For ease of exposton, we wll focus on the case n whch each demand s equal to 1. However, t s a relatvely straghtforward exercse to extend the algorthm and ts analyss to the case of general demands. We wll use the terms assgnment cost and servce cost nterchangeably. Our Technques. The outlne of our algorthm s as follows. Gven the optmal LP soluton and ts dual, we vew the optmal prmal soluton as a bpartte graph n whch the nodes correspond to faclty locatons and clents, and the

3 edges correspond to pars (, j) such that a postve fracton of the demand at clent j s assgned to faclty by the LP soluton. We use ths to construct a partton of the demand and facltes nto clusters: each cluster s centered at a clent, and the neghbors of ths clent contaned n the cluster are opened (n the fractonal soluton) n total at least 1/2. Each fractonally open faclty locaton wll, ultmately, be assgned to some cluster (.e., not every faclty assgned to ths cluster need be a neghbor of the center), and each cluster wll be expected to serve all of the demand that ts facltes serve n the fractonal soluton. Each faclty that s fully opened n the fractonal soluton can mmedately be opened and serve all of ts demand; we vew the remanng demand as located at the cluster center, and fnd a soluton to the sngle-demand capactated faclty locaton problem nduced by ths cluster to determne the other facltes to open wthn ths cluster. Pecng ths together for each cluster, we then solve a transportaton problem to determne the correspondng assgnment. To analyze ths procedure, we show that the LP soluton can also be decomposed nto feasble fractonal solutons to the respectve sngle-demand problems. Our algorthm for the sngle-node subproblems computes a roundng of ths fractonal soluton, and t s mportant that we can bound the ncrease n cost ncurred by ths roundng. Furthermore, note that t wll be mportant for the analyss (and the effectveness of the algorthm) that we ensure that n movng demand to a cluster center, we are not movng t too much, snce otherwse the soluton created for the sngle-node problem wll be prohbtvely expensve for the true locaton of the demand. One novel aspect of our analyss s that the performance guarantee analyss comes n two parts: a part that s related to the fact that the assgnment costs are ncreased by ths dsplacement of the demand, and a part that s due to the aggregated effect of roundng the fractonal solutons to the sngle-node problems. One consequence of ths s that our analyss s not the clent-by-clent analyss that has become the domnant paradgm n the recent flurry of work n ths area. Fnally, our analyss reles on both the prmal and dual LPs to bound the cost of the soluton computed. In dong ths, one sgnfcant dffculty s that the terms n the dual objectve that correspond to the upper bound for the hard capacty have a -1 as ther coeffcent; however, we show that further structure n the optmal prmal-dual par that results from the complementary slackness condtons s suffcent to overcome ths obstacle (n a way smlar to that used earler n [12]). Although our analyss apples only to the case n whch the fxed costs are equal, our algorthm s suffcently general to handle arbtrary fxed costs. Furthermore, we beleve that our approach may prove to be a useful frst step n analyzng more sophstcated LP relaxatons of the capactated faclty locaton problem; n partcular, we beleve that the decomposton nto sngle-node problems can be a provable effectve approach n the more general case. Specfcally, we conjecture that the extended flow cover nequaltes of Padberg, Van Roy, and Wolsey [7] as adapted by Aardal [1] are suffcent to nsure a constant ntegralty gap; ths rases the possblty of buldng on a recent result of Carr, Flescher,

4 Leung, and Phllps [2] that showed an analogous result for the sngle-demand node problem. 2 A Lnear Program We can formulate the capactated faclty locaton problem as an nteger program and relax the ntegralty constrants to get a lnear program (LP). We use to ndex the facltes n F and j to ndex the clents n D. mn s.t. f y + d j c j x j (P) j x j 1 j (1) x j y, j (2) d j x j u y (3) j y 1 (4) x j,y 0, j. Varable y ndcates f faclty s open and x j ndcates the fracton of the demand of clent j that s assgned to faclty. The frst constrant states that each clent must be assgned to a faclty. The second constrant says that f clent j s assgned to faclty then must be open, and constrant (3) says that at most u amount of demand may be assgned to. Fnally (4) says that a faclty can only be opened once. A soluton where the y varables are 0 or 1 corresponds exactly to a soluton to our problem. The dual program s, max j α j z (D) s.t. α j d j c j + β j + d j γ, j (5) β j f + z u γ (6) j α j,β j,γ,z 0, j. Intutvely α j s the budget that j s wllng to spend to get tself assgned to an open faclty. Constrant (5) says that a part of ths s used to pay for the assgnment cost d j c j and the rest s used to (partally) pay for the faclty openng cost. For convenence, n what follows, we consder unt demands,.e., d j =1for all j. The prmal constrant (3) and the dual constrant (5) then smplfy to, j x j u y,andα j c j + β j + γ, and the objectve functon of the prmal program (P) s mn f y + j, c jx j. All our results contnue to hold n the presence of arbtrary demands d j f the demand of a clent s allowed to be assgned to multple facltes.

5 3 Roundng the LP In ths secton we gve a 5-approxmaton algorthm for capactated faclty locaton when all faclty costs are equal. We wll round the optmal soluton to (P) to an nteger soluton losng a factor of at most 5, thus obtanng a 5-approxmaton algorthm. 3.1 The Sngle-Demand-Node Capactated Faclty Locaton Problem The specal case of capactated faclty locaton where we have just one clent or demand node (called SNCFL) plays an mportant role n our roundng algorthm. Ths s also known as the sngle-node fxed-charge problem [7] or the sngle-node capactated flow problem. The lnear program (P) smplfes to the followng. mn s.t. f v + c w (SN-P) w D w u v (7) v 1 (8) w,v 0. Here D s the total demand that has to be assgned, f 0 s the fxed cost of faclty, andc 0 s the per unt cost of sendng flow, or dstance, to faclty. Varable w s the total demand (or flow) assgned to faclty, andv ndcates f faclty s open. We show that a smple greedy algorthm returns an optmal soluton to (SN-P) that has the property that at most one faclty s fractonally open,.e., there s at most one such that 0 <v < 1. We wll explot ths fact n our roundng scheme. Gven any feasble soluton (w, v) wecansetˆv = w u and obtan a feasble soluton (w, ˆv) of no greater cost. So we can elmnate the v varables from (SN-P), changng the objectve functon to mn f ) ( u + c w, and replacng constrants (7), (8) by w u for each. Clearly, ths s equvalent to the earler formulaton. It s easy to see now that the followng greedy algorthm delvers an optmal soluton: start wth w = v =0forall. Consder facltes n ncreasng order of f u +c value and assgn to faclty a demand equal to u or the resdual demand left, whchever s smaller,.e., set w =mn(u, demand left), v = w u, untl all D unts of demand have been assgned. We get the followng lemma. Lemma 3.1. The greedy algorthm that assgns demand to facltes n ncreasng order of f u + c delvers an optmal soluton to (SN-P). Furthermore,there s at most one faclty n the optmal soluton such that 0 <v < 1.

6 3.2 The Algorthm We now descrbe the full roundng procedure. Let (x, y) and(α, β, γ, z) bethe optmal solutons to (P) and (D) respectvely, and OPT be the common optmal value. We may assume wthout loss of generalty that x j = 1 for every clent j. We frst gve an overvew of the algorthm. Our algorthm runs n two phases. In the frst phase, we partton the facltes such that y > 0ntoclusters each of whch wll be centered around a clent that we wll call the cluster center. We denote the cluster centered around clent k by N k. The cluster N k conssts of ts center k, the set of facltes assgned to t, and the fractonal demand served by the these facltes,.e., N k j x j. The clusterng phase mantans two propertes that wll be essental for the analyss. It ensures that, (1) each cluster contans a fractonal faclty weght of at least 1 2,.e., N k y 1 2, and (2) f some faclty n cluster N k fractonally serves a clent j, then the center k s not too far away from j (we make ths precse n the analyss). To mantan the second property we requre a somewhat more nvolved clusterng procedure than the one n [11]. In the second phase of the algorthm we decde whch facltes wll be (fully) opened n each cluster. We consder each cluster separately, and open enough facltes n N k to serve the fractonal demand assocated wth the cluster. Ths s done n two steps. Frst, we open every faclty n N k wth y =1.Next,wesetupannstanceofSNCFL. The nstance conssts of all the remanng facltes, and the entre demand served j x j, consdered as concentrated at the by these facltes, D k = N k :y <1 center k. Now we use the greedy algorthm above to obtan an optmal soluton to ths nstance wth the property that at most one faclty s fractonally open. Snce the faclty costs are all equal and each cluster has enough faclty weght, we can fully open ths faclty and charge ths aganst the cost that the LP ncurs n openng facltes from N k. Pecng together the solutons for the dfferent clusters gves a soluton to the capactated faclty locaton nstance, n whch every faclty s ether fully open or closed. Now we compute the mn-cost assgnment of clents to open facltes by solvng a transportaton problem. We now descrbe the algorthm n detal. Let F = { : y > 0} be the (partally) opened facltes n (x, y), and F j = { : x j > 0} be the facltes n F that fractonally serve clent j. 1. Clusterng. Ths s done n two steps. C1. At any stage, let C be the set of the current cluster centers, whch s ntally empty. We use N k to denote a cluster centered around clent k C. For each clent j/ C, we mantan a set B j of unclustered facltes that are closer to t than to any cluster center,.e., B j = { F j : / k C N k and c j mn k C c k }. (Ths defnton of B j s crucal n our analyss that shows that f clent j s fractonally served by N k,then k s not too far from j.) We also have a set S contanng all clents that could be chosen as cluster centers. These are all clents j/ Cthat send at least half of ther demand to facltes n B j,.e., S = {j / C: B j x j 1 2 }. Of course, ntally S = D, sncec =.

7 Whle S s not empty, we repeatedly pck j Swth smallest α j value and form the cluster N j = B j around t. We update the sets C and S accordngly. (Note that for any cluster N k,wehavethat N k y N k x k 1 2.) C2. After the prevous step, there could stll be facltes that are not assgned to any cluster. We now assgn these facltes n U = F k C N k to clusters. We assgn U to the cluster whose center s nearest to t,.e., we set N j = N j {} where j =argmn k C c k. In addton, we assgn to the cluster all of the fractonal demand served by faclty. (Afterths step, the clusters N j,j Cpartton the set of facltes F.) 2. Reducng to the sngle-node nstances. For each cluster N k,wefrst open each faclty n N k wth y = 1. We now create an nstance of SNCFL on the remanng set of facltes, by consderng the total demand assgned to these facltes as beng concentrated at the cluster center k. Sooursetof facltes s L k = { N k : y < 1}, eachc s the dstance c k, and the total demand s D k = L k j x j. We use the greedy algorthm of Secton 3.1 to fnd an optmal soluton (w (k),v (k) ) to ths lnear program. Let Ok be the value of ths soluton. We call the faclty such that 0 <w (k) < 1(f such a faclty exsts) the extra faclty n cluster N k. We fully open all the facltes n L k wth w (k) > 0 (ncludng the extra faclty). Note that the facltes opened (ncludng each such that y = 1) have enough capacty to satsfy all the demand N k j x j. Pecng together the solutons for all the clusters, we get a soluton where all the y varables are assgned values n {0, 1}. 3. Assgnng clents. We compute a mnmum cost assgnment of clents to open facltes by solvng the correspondng transportaton problem. 3.3 Analyss The performance guarantee of our algorthm wll follow from the fact that the decomposton constructed by the algorthm of the orgnal problem nstance nto sngle-node subproblems, one for each cluster, satsfes the followng two nce propertes. Frst, n Lemma 3.5, we show that the total cost of the optmal solutons for each of these sngle-node nstances s not too large compared to OPT. We prove ths by showng that the LP soluton nduces a feasble soluton to (SN-P) for the SNCFL nstance of each cluster and that the total cost of these feasble solutons s bounded. Second, n Lemma 3.7, we show that the optmal solutons to each of these sngle-node nstances obtaned by our greedy algorthm n Secton 3.1, can be mapped back to yeld a soluton to the orgnal problem n whch every faclty s ether opened fully, or not opened at all, whle losng a small addtve term. Pecng together these partal solutons, we construct a soluton to the capactated faclty locaton problem. The cost of ths soluton s bounded by aggregatng the bounds obtaned for each partal soluton. We note that ths bound s not based on a clent-by-clent analyss, but rather on boundng the cost generated by the overall cluster.

8 Observe that there are two sources for the extra cost nvolved n mappng the solutons to the sngle-node nstances. We mght need to open one fractonally open faclty n the optmal fractonal soluton to (SN-P). Ths s bounded n Lemma 3.6, and ths s the only place n the entre proof whch uses the assumpton that the fxed costs are all equal. In addton, we need to transfer all of the fractonal demand that was assumed to be concentrated at the center of the cluster, back to ts orgnal locaton. To bound the extra assgnment cost nvolved, we rely on the mportant fact that f a clent j s fractonally served by some faclty N k, then the dstance c jk s bounded. Snce the trangle nequalty mples that c jk c j + c k, we focus on boundng the dstance c k. Ths s done n Lemmas 3.3 and 3.4. In Lemma 3.8, we provde a bound on the faclty cost and assgnment cost nvolved n openng the facltes wth y =1, whch, by relyng on complementary slackness, overcomes the dffcultes posed by the z term n the dual objectve functon. We then combne these bounds to prove our man theorem, Theorem 3.9, whch states that the resultng feasble soluton for the capactated faclty locaton problem s of cost at most 5 OPT. We frst prove the followng lemma that states a necessary condton for a faclty to be assgned to cluster N k. Lemma 3.2. Let be a faclty assgned to cluster N k n step C1 or C2. Let C be the set of cluster centers just after ths assgnment. Then, k s the cluster center closest to among all cluster centers n C ; that s, c k =mn k C c k. Proof. Snce k C, clearly we have that c k mn k C c k.if s assgned n step C1, then t must be ncluded when the cluster centered at k s frst formed; that s, B k and the lemma holds by the defnton of B k.otherwse,f s assgned n step C2, then C s the set of all cluster centers, n whch case t s agan true by defnton. For a clent j, consder the pont when j was removed from the set S n step C1, ether because a cluster was created around t, or because the weght of the facltes n B j decreased below 1 2 when some other cluster was created. Let A j = F j \ B j be the set of facltes not n B j at that pont. Recall that there are two reasons for removng a faclty from the set B j : t was assgned to some cluster N k, or there was some cluster center k C, such that c k <c j.we defne (j) as the faclty n A j nearest to j. Also, observe that once j/ C S, then we have that A j x j > 1 2. Lemma 3.3. Consder any clent j and any faclty A j.if s assgned to cluster N k,thenc k α j. Proof. If k = j, (j could be a cluster center), then we are done snce A j F j and x j > 0 mples that c j α j (by complementary slackness). Otherwse, consder the pont when j was removed from S n step C1, and let C be the set of cluster centers just after j s removed. Note that j could belong to C f t s a cluster center. Snce / B j at ths pont, ether N k for some k C or

9 we have that c j > mn k C {j} c k. In the former case, t must be that k = k, snce the clusters are dsjont. Also, c k α k,sncen k F k,andα k α j,snce k was pcked before j from S (recall the order n whch we consder clents n S). In the latter case, consder the set of cluster centers C just after s assgned to N k (ether n step C1 or step C2), and so k C. It must be that C C, snce was removed from B j before t was assgned to N k, and by Lemma 3.2, c k =mn k C c k. Hence, c k mn k C {j} c k <c j α j snce A j F j. Lemma 3.4. Consder any clent j and a faclty F j \ A j.let be assgned to cluster N k.ifj C,thenc k c j ;otherwse,c k c j + c (j)j + α j. Proof. If j s a cluster center, then when t was removed from S, wehaveconstructed the cluster N j equal to the current set B j, whch s precsely F j \ A j. So s assgned to N j,thats,k = j, and hence the bound holds. Suppose j / C. Consder the pont just before the faclty (j) s removed from the set B j n step C1, and let C be the set of cluster centers at ths pont. By the defnton of the set A j, j s stll a canddate cluster center at ths pont. Let k C be the cluster center due to whch (j) was removed from B j,andso ether (j) N k F k or c (j)k <c (j)j. In each case, we have c (j)k α j, snce the choce of k mples that α k α j. Now consder the set of cluster centers C just after s assgned to N k.snce/ A j, (j) was removed from B j before ths pont. So we have C C. Usng Lemma 3.2, c k = mn k C c k c k c j + c (j)j + c (j)k c j + c (j)j + α j. Consder now any cluster N k. Recall that L k = { N k : y < 1}, (w (k),v (k) ) s the optmal soluton to (SN-P) found by the greedy algorthm for the snglenode nstance correspondng to ths cluster, and Ok s the value of ths soluton. Let k() Cdenote the cluster to whch faclty s assgned, and so N k(). Lemma 3.5. The optmal value Ok L k f y + j, L k c k x j, and hence, k C O k :y f <1 y + j,:y c <1 k()x j. Proof. The second bound follows from the frst snce the clusters N k are dsjont. We wll upper bound Ok by exhbtng a feasble soluton (ŵ, ˆv) ofcostatmost the clamed value. Set ˆv = y,andŵ = j x j for all L k.notethat ŵ = L k j x j = D k. The faclty cost of ths soluton s at most L k f ˆv = L k f y.theservcecosts L k c ŵ = j, L k c k x j. Combnng ths wth the bound on faclty cost proves the lemma. Lemma 3.6. The cost of openng the (at most one) extra faclty n cluster N k s at most 2 N k f y. Proof. We have N k y N k x k 1 2 snce N k was created n step C1 and s centered around k, and no faclty s removed from N k n step C2. We open at most one extra faclty from N k. Snce all facltes have the same cost f, the cost of openng ths faclty s f f 2 N k y =2 N k f y. Ths s the only place where we use the fact that the faclty costs are all equal.

10 Let ŷ be the 0-1 vector ndcatng whch facltes are open,.e., ŷ =1f s open, and 0 otherwse. We let ŷ (k) denote the porton of ŷ consstng of the facltes n L k,.e., ŷ (k) = ( ŷ (k) and ŷ ) L (k) k =1f L k s open, and 0 otherwse. Lemma 3.7. The soluton ( w (k),v (k)) for cluster N k yelds an assgnment ˆx (k) = ) such that, (ˆx (k) j L k,j D () (ˆx (k), ŷ (k) ) obeys constrants (2) (4) for all L k, () ˆx satsfes L k x j fracton of the demand of each clent j, that s, L k ˆx j = L k x j for all j and, () the cost L k f ŷ (k) + j, L k c j ˆx (k) j s at most Ok +2 N k f y + j, L k c j x j + j, L k c k x j. +c w (k) ). Constrants (4) are clearly satsfed s at most Proof. We have Ok = ( L f k v (k) for L k,snceŷ (k) s a {0, 1}-vector. The faclty cost L k f ŷ (k) L k f v (k) +2 N k f y snce every faclty other than the extra faclty s ether fully open or not open n the soluton (w (k),v (k) ) and the cost of openng the extra faclty s at most 2 N k f y by Lemma 3.6. We set the varables ˆx (k) j can be bounded by L k c w (k) for L k so that the servce cost j, L k c j ˆx (k) j + j, L k (c j +c k )x j. Combnng ths wth the above bound on the faclty cost, proves the lemma. The servce cost of the snglenode soluton s the cost of transportng the entre demand D k = j, L k x j from the facltes n L k to the center k, and now we want to move the demand, L k x j, of clent j from k back to j. Dong ths for every clent j ncurs an addtonal cost of j L k c jk x j j, L k (c j + c k )x j. More precsely, we set ˆx (k) j, L k arbtrarly so that, (1) L k ˆx (k) j = L k x j for every clent j, and (2) j ˆx(k) j = w (k) for every faclty L k. Ths satsfes constrants (2),(3) fˆx (k) j > 0thenw (k) > 0, so ŷ (k) =1,and j ˆx(k) j = w (k) u = u ŷ (k).the servce cost s, c j ˆx (k) j c k ˆx (k) j + c jkˆx (k) j c w (k) + (c j + c k )x j. j, L k L k,j j, L k L k j, L k Lemma 3.8. The cost of openng facltes wth y =1,andforeachsuch, of sendng x j unts of flow from j to for every clent j, satmost j,:y =1 α jx j z. Proof. Ths follows from complementary slackness. Each faclty wth z > 0 has y = 1. For any such faclty we have,

11 α j x j j = j = j = j c j x j + j c j x j + j c j x j + f + z. β j x j + j γ x j ( xj > 0 α j = c j + β j + γ ) ( ) βj > 0 x j = y, β j y + u γ y γ > 0 j x j = u y ( y > 0 j β ) j + u γ = f + z Summng over all wth y = 1 proves the lemma. Puttng the varous peces together, we get the followng theorem. Theorem 3.9. The cost of the soluton returned s at most 5 OPT. Proof. To bound the total cost, t suffces to gve a fractonal assgnment (ˆx j ) such that (ˆx, ŷ) s a feasble soluton to (P) and has cost at most 5 OPT. We construct the fractonal assgnment as follows. Frst we set ˆx j = x j for every faclty wth y =1=ŷ. Ths satsfes constrants (2) (4) for such that y =1. By the prevous lemma we have, f ŷ + c j ˆx j = z. (9) :y =1 j,:y =1 j,:y =1 α j x j Now for each cluster N k,wesetˆx j =ˆx (k) j for L k where (ˆx (k), ŷ (k) )sthe partal soluton for cluster N k gven by Lemma 3.7. All other ˆx j varables are 0. Applyng parts () and () of Lemma 3.7 for all k C,wegetthat(ˆx, ŷ) satsfes (2) (4) for every such that y < 1, and :y ˆx <1 j = :y x <1 j for every clent j Hence, (ˆx, ŷ) satsfes constrants (2) (4) and ˆx j = :y x =1 j + :y x <1 j = 1, showng that (ˆx, ŷ) s a feasble soluton to (P). Snce the clusters N k are dsjont, from part () of Lemma 3.7, we have, f ŷ + f y + c j x j + c k() x j :y <1 c j ˆx j Ok +2 k C 3 f y + c j x j +2 c k() x j. where the last nequalty follows from Lemma 3.5. For any clent j and faclty F j,f A j,thenwehavec k() α j by Lemma 3.3; otherwse, by Lemma 3.4, c k() c j c j + α j for j C,andc k() c j + c (j)j + α j for j/ C. Pluggng ths n the above expresson we get, f ŷ + f y + c j x j +2 α j x j :y <1 c j ˆx j 3 +2 j c jx j + :y <1 / A j j/ C 2c (j)j :y <1 / A j x j.

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