Facty Locaton wth Servce Instaaton Costs (Extended Abstract) Davd B. Shmoys Chatanya Swamy Retsef Lev Abstract We consder a generazaton of the uncapactated facty ocaton probem whch we ca Facty Locaton wth Servce Instaaton Costs. We are gven a set of factes, F, a set of demands or cents D, and a set of servces S. Each facty has a facty openng cost f, and we have a servce nstaaton cost of f for every facty-servce par (, ). Each cent j n D requests a specfc servce g(j) Sand the cost of assgnng a cent j to facty s gven by c j. We want to open a set of factes, nsta servces at the open factes, and assgn each cent j to an open facty at whch servce g(j) s nstaed, so as to mnmze the sum of the facty openng costs, the servce nstaaton costs and the cent assgnment costs. Our man resut s a prma-dua 6-approxmaton agorthm under the assumpton that there s an orderng on the factes such that f comes before n ths orderng then for every servce type, f f. Ths ncudes (as speca cases) the settngs where the servce nstaaton cost f depends ony on the servce type, or depends ony on the ocaton. Wth arbtrary servce nstaaton costs, the probem becomes as hard as the set-cover probem. Our agorthm extends the agorthm of Jan & Vazran [9] n a nove way. If the servce nstaaton cost depends ony on the servce type and not on the ocaton, we gve an LP roundng agorthm that attans an mproved approxmaton rato of 2.391. The agorthm combnes both custered randomzed roundng [6] and the fterng based technque of [10, 14]. We aso consder the k-medan verson of the probem where there s an addtona requrement that at most k factes may be opened. We use our prma-dua agorthm to gve a constant-factor approxmaton for ths probem when the servce nstaaton cost depends ony on the servce type. 1 Introducton Facty ocaton probems have been wdey studed n the Operatons Research communty (see for e.g. [12]). In ts smpest verson, uncapactated facty ocaton (UFL), we are gven a set of factes, F, and a set of demands or cents D. Each facty has a facty openng cost f and the cost of assgnng a cent j to facty s gven by c j.wewanttoopen some factes {shmoys,swamy}@cs.corne.edu. Dept. of Computer Scence, Corne Unversty, Ithaca, NY 14853. Research supported partay by NSF grant CCR-9912422. r227@corne.edu. Schoo of ORIE, Corne Unversty, Ithaca, NY 14853. Research supported partay by a grant from Motoroa and NSF grant CCR-9912422. from the set F and assgn each demand to an open facty. The goa s to mnmze the sum of the facty openng costs and the cent assgnment costs. Ths probem has a wde range of appcatons. For exampe, a company mght want to open ts warehouses at some ocatons so that ts tota cost of openng warehouses and servcng customers s mnmzed. In varous appcatons, the cents are dfferentated accordng to the knd of servce they requre and to satsfy the servce requrement of a cent we have to assgn t to a facty that can provde the servce requred by the cent. For exampe, n the warehouse ocaton probem above, the customers may be reta stores that request dfferent knds of suppes. A warehouse may store dfferent knds of suppes. To satsfy a customer we have to assgn t to a warehouse that hods nventory of the type requested by the customer. We mode such a settng by sayng that n addton to factes and cents, we have a set of servces S. Each cent j n D requests a specfc servce g(j) S. To satsfy cent j we have to assgn t to an open facty on whch servce g(j) s nstaed. Further, f we nsta servce on an open facty we ncur a servce nstaaton cost of f. We want to open a set of factes, nsta servces at the open factes, and assgn each cent j to an open facty such that servce g(j) s nstaed at. The cost of a souton s the sum of the facty openng costs, the servce nstaaton costs and the cent assgnment costs, and the goa s to fnd a souton wth mnmum tota cost. We ca ths probem, Facty Locaton wth Servce Instaaton Costs. In the warehouse ocaton probem, the servce nstaaton cost corresponds to the nta cost of settng up the warehouse to store the partcuar knd of nventory. The noton of servce-dependent fxed costs s aso used n nventory probems where one ncurs a jont setup cost to start a new order and an tem-dependent fxed cost to order a specfc tem, so one needs to coordnate the pacement of tem orders; see [1] for a survey. We assume throughout that the assgnment costs c j form a metrc. Copyrght 2004 by the Assocaton for Computng Machnery, Inc. and the Socety for ndustra and Apped Mathematcs. A Rghts reserved. Prnted n The Unted States of Amerca. No part of ths book may be reproduced, stored, or transmtted n any manner wthout the wrtten permsson of the pubsher. For nformaton, wrte to the Assocaton for Computng Machnery, 1515 Broadway, New York, NY 10036 and the Socety for Industra and Apped Mathematcs, 3600 Unversty Cty Scence Center, Phadepha, PA 19104-2688 1088
Appcatons and Reated Work. Facty ocaton wth servce nstaaton costs can be used to mode a data management/cachng probem. Here we are gven a set of ocatons n a network at whch caches may be but and a set of processes ocated at the nodes of the network that request data tems. Each process requests a specfc data tem. To satsfy the request we must assgn the process to a cache that stores the requested data tem, ncurrng an access cost proportona to the dstance between the process ste and the cache ocaton. Budng a cache at a ocaton ncurs a ocaton dependent cost and storng a data tem n a cache at a partcuar ocaton ncurs a cost that depends on the data tem and the ocaton. The goa s to bud caches, store data tems n the caches and assgn each process to a cache contanng the data tem requested by the process, so as to mnmze the tota cost of budng caches, storng data tems and the access cost of requests. Ths s exacty the facty ocaton probem wth servce nstaaton costs where the caches are factes, the processes are cents and the data tems correspond to servces. Baev & Rajaraman [3] consdered a cosey reated probem caed the data pacement probem and gave a 20.5- approxmaton agorthm. Here caches of fxed capacty are aready but at certan ocatons and the goa s to fnd a pacement of data tems to caches respectng the cache capactes that mnmzes the sum of the access costs and the cost of storng data tems. The rato has recenty been mproved by Swamy [16]. Facty ocaton wth servce nstaaton costs s a generazaton of UFL f there s just one servce type then ths s smpy the uncapactated facty ocaton probem. There s a arge body of terature that deas wth desgnng approxmaton agorthms for metrc UFL and we sampe ony a few resuts beow; see [13] for a survey of ths and earer work. Shmoys, Tardos & Aarda [14] gave the frst constantfactor approxmaton agorthm for ths probem usng the fterng technque of Ln & Vtter [10] to round the optma souton of a near program. Chudak & Shmoys [5, 6] gave an LP roundng based ( ) 1+ 2 e - approxmaton agorthm. They combned randomzed roundng and the decomposton resuts of [14] to get a varant that mght be caed custered randomzed roundng. Svrdenko [15] mproved the rato to 1.58. Jan & Vazran [9] gave a combnatora prma-dua 3-approxmaton agorthm where the LP s used ony n the anayss. The current best rato for UFL s 1.52 [11] obtaned by budng upon a dua-fttng based greedy agorthm of Jan, Mahdan, Markaks, Saber & Vazran [8]. Our Resuts. Our man resut s a prma-dua 6- approxmaton agorthm for the facty ocaton probem wth servce nstaaton costs under the assumpton that there s an orderng on the factes such that f comes before n ths orderng then for every servce type, f f. Ths s reasonabe n many settngs; for exampe, one expects the nventory setup cost of a warehouse n New York cty to be ess than the nventory setup cost n a remote town ke Ithaca regardess of the knd of nventory. As speca cases ths ncudes the cases where the servce nstaaton cost f depends ony on the servce type, or depends ony on the ocaton. In the former settng where the servce nstaaton cost depends ony on the servce type we gve an agorthm based on LP roundng that attans a much mproved approxmaton rato of 2.391. We show that wth arbtrary servce nstaaton costs the probem becomes as hard as the set-cover probem. Combned wth the resut of Fege [7], ths shows that no poynoma-tme agorthm wth a rato of (1 ɛ)n D exsts for ths probem n the genera case uness NP DTIME[n O(og og n) ]. We aso consder the k-medan verson of the probem where there s an addtona requrement that at most k factes may be opened. We use our prma-dua agorthm to gve a constant-factor approxmaton for ths probem when the servce nstaaton cost depends ony on the servce type. Our Technques. Facty ocaton wth servce nstaaton costs s a generazaton of UFL. It dffers however from tradtona mut-eve extensons of facty ocaton where we assume that a demand can be assgned to any facty n any eve. In our probem demand j may ony be assgned to a eve 1 facty (, g(j)) and then to facty n eve 2. Moreover, exstng technques for UFL and mut-eve facty ocaton do not ready generaze. If there were no facty openng costs we coud decoupe the probem nto severa UFL nstances, one for each servce type, and sove each one separatey. Wth facty openng costs ths approach fares bady snce we may end up openng a ot of factes and spend too much on the facty openng costs. A known agorthms for UFL rey on the fact, ether n the desgn or the anayss, that a cent j can be moved from a facty to another nearby facty wthout ncreasng ts assgnment cost by much, and eavng the facty openng cost unchanged. In our probem, reassgnng j to may now requre us to nsta servce g(j) on causng us to pay the nstaaton cost f g(j) whch coud be arge. Techncay the hard part s to fnd a way to reassgn cents to nearby factes so that we do not pay too much to nsta servces at the new ocatons. Wth arbtrary servce nstaaton costs such a reassgnment 1089
need not be possbe snce we can encode the constrant that a cent may ony be assgned to a specfc set of factes, makng the probem set-cover hard. We bud upon the prma-dua agorthm of [9] n a nove way and gve a 6-approxmaton agorthm under the assumpton that the factes are ordered so that f comes before then f f for every servce. At a hgh eve, the dea s to consder an nteger programmng formuaton of the probem and the dua of ts near programmng reaxaton, and construct smutaneousy an nteger prma souton and a dua souton. Each cent j has a dua varabe α j whch can be nterpreted ntutvey as the payment that j s wng to make to get tsef assgned to an open facty. The Jan-Vazran (JV) agorthm [9] for UFL works n two phases. In phase I we grow each dua varabe α j unformy and graduay bud a prma feasbe souton. Once α j becomes equa to c j for some facty, j starts payng toward the facty openng cost of. When the tota contrbuton to from the varous cents equas f, we decare to be tentatvey open and assgn a the unassgned cents contrbutng toward to. PhaseI ends when each cent s assgned to a tentatvey open facty. At ths pont a cent coud be contrbutng towards mutpe tentatvey open factes. We ca a set of factes ndependent f each cent contrbutes towards at most one facty n the set. In phase II we seect a maxma ndependent subset of tentatvey open factes and open these. The anayss shows that f the facty to whch a cent j was assgned n phase I s not opened, then there s a nearby open facty to whch j can be reassgned. Our agorthm aso proceeds n phases. Durng phase I we tentatvey open some factes, tentatvey nsta servce on some factes for each servce type, and assgn each cent j to a tentatvey open facty on whch servce g(j) s tentatvey nstaed. Phase II s more nvoved. We have to seect a set of factes to open and nsta servces n such a way that we can, (1) pay for nstang the servces, and (2) ensure that f a cent j has to be reassgned there s a nearby open facty on whch servce g(j) s nstaed. We show that we can acheve propertes (1) and (2) f we ook at the factes n a partcuar order and pck a maxma ndependent subset greedy. Ths gves us a 6-approxmaton agorthm. Our prma-dua agorthm expots the property that n the JV agorthm any maxma ndependent set of tentatvey open ocatons may be pcked. Athough ths s a we-known fact, to our knowedge ths has been used n ony a coupe of appcatons prevousy. Barta, Charkar and Raz [4] consder a custerng probem and use the JV agorthm to sove a reaxaton of the probem by pckng an approprate maxma ndependent set. Archer, Rajagopaan and Shmoys [2] pck a maxmum ndependent set and use ths to prove a bound on the ntegraty gap of the k-medan LP. When the servce nstaaton cost depends ony on the servce type, we gve an LP-roundng agorthm that combnes custered randomzed roundng [6] and the fterng based technque of [10, 14]. A feature of the agorthm s that we bound dstances c j usng both the α j bound due to compementary sackness and the bound obtaned by fterng. Ths gves a better performance guarantee than that obtaned by usng ether of the two bounds separatey. Svrdenko [15] aso used the two bounds n conjuncton to mprove the approxmaton rato for UFL from ( ) 1+ 2 e to 1.58. 2 A Lnear Program We can formuate the probem as an nteger program and reax the ntegraty constrants to get a near program. We use to ndex the factes n F, j to ndex the cents n D and to ndex the servces n S. mn f y + f y + c j x j (P) j s.t. x j 1 j x j y g(j), j x j y, j x j,y,y 0, j,. Varabe y ndcates f facty s open, y ndcates f servce type s nstaed at, andx j ndcates f cent j s connected to facty. The frst constrant states that each cent must be assgned to a facty, the second and the thrd constrants say that f cent j s assgned to facty, then servce g(j) mustbenstaedonand must be open. An ntegra souton corresponds exacty to a souton to our probem. Let G be the set of cents requestng servce. The dua program s, max j α j (D) s.t. α j c j + β j + θ j, j (2.1) θ j f, j G β j f (2.2) j α j,β j,θ j 0, j. Intutvey α j s the budget that j s wng to spend to get tsef assgned to an open facty. Constrant (2.1) says that a part of ths goes towards payng for 1090
the assgnment cost c j. The rest gets dvded nto a payment for the servce nstaaton cost θ j,anda payment for the facty openng cost β j. 3 A Prma-Dua Agorthm We consder nstances of the probem where there s an orderng on the factes n F such that f comes before n ths orderng then for every servce type, f f. Equvaenty ths means that for any two ocatons,, the vectors ( ) f T =1... S and ( ) f T are comparabe. =1... S Let O denote ths tota orderng on the factes. We say that f comes before n the orderng O. The agorthm s strongy motvated by the prmadua agorthm of Jan and Vazran for the tradtona uncapactated facty ocaton agorthm. In our agorthm, we frst construct a feasbe dua souton, and then use ths dua souton to extract a feasbe (nteger) prma souton. As has been the norm, our agorthm s a dua ascent agorthm, so a dua varabes are ony ncreased throughout the executon of the agorthm. We next descrbe the agorthm. There s a noton of tme around whch the agorthm s specfed. We start at tme t = 0, a dua varabes are ntazed to 0, each demand j s sad to be unfrozen, anda factes are cosed. At frst, the varabes that are ncreased are the α j s; more precsey, for any unfrozen demand j, α j s aways equa to the tme t. We say that demand j s tght wth facty, or has reached, f α j c j. As tme ncreases, we w freeze demand ponts, tentatvey open factes, and tentatvey nsta servces at factes. We ncrease the α j of each demand j unt one of the foowng events happens: 1. Suppose that demand j becomes tght wth facty. If servce g(j) s not tentatvey nstaed at, then we start ncreasng θ j at the same rate as α j ; that s, f α j = t, thenθ j = t c j. If servce g(j) s tentatvey nstaed, but s not tentatvey open, we nstead ncrease β j at the same rate as α j ;that s, f α j = t, thenθ j remans 0, but β j = t c j. Fnay, f servce g(j) s tentatvey nstaed, and s tentatvey open, we freeze demand pont j (and no onger ncrease α j ). 2. Suppose that for a facty and a servce type, we get that j G θ j = f : n ths case, we tentatvey nsta servce at. If s aso tentatvey open, then we freeze each demand j G that s tght wth (and no onger ncrease α j ). If s not yet tentatvey open, then for each demand j G that s tght wth, we no onger ncrease θ j, but nstead start ncreasng β j at the same rate as α j. 3. Suppose that for a facty, j β j = f : n ths case, we tentatvey open. For each demand j, we do not ncrease β j from now on. If demand j s tght wth and servce g(j) s tentatvey nstaed at, we freeze j (and no onger ncrease α j ). We ony rase the α j,β j,θ j of unfrozen demands. Frozen demands do not partcpate n any events. We contnue ths process unt a demands become frozen. Let (α, β, θ) denote the fna dua souton obtaned by the above process. Observe that f s the facty that caused j to freeze, then servce g(j) must be tentatvey nstaed at, and must be tentatvey open. We now specfy whch factes to open, how to nsta servces on factes, and how to assgn demands to factes. Let F be the set of tentatvey open factes, and et F F be the set of tentatvey open factes on whch servce s tentatvey nstaed. For facty F, et t be the tme at whch became tentatvey open. If F,ett be the tme at whch servce was tentatvey nstaed at. Openng factes. We open a subset of factes from F. We say that, F are dependent f there s a demand j such that both β j and β j are postve. We consder the factes n F n the order gven by O and pck a maxma ndependent set of factes, F F. We open the factes n F. Instang servces. Consder servce type and the set of factes F. We say that factes, F are servce--dependent f there exsts some demand j n G such that both θ j and θ j are postve. We pck a maxma ndependent subset F by ookng at factes n F n a partcuar order: frst we consder factes n F F n ncreasng order of t, and then factes n F F n ncreasng order of t. We frst nsta servce on a factes n F F. Furthermore, for each F F, we pck a facty F such that and are dependent and (n the orderng gven by O), and nsta servce on facty.wesaythat s the neghbor of and denote t by nbr(). Note that nbr(.) depends ony on and not on the servce type : f/ F, then we can choose a snge facty F such that regardess of the servce type. Assgnng demands. We assgn each cent j to the nearest open facty at whch servce g(j) s nstaed. 3.1 Anayss. We now bound the performance of our agorthm. The foowng emma just says what t means for a demand j to get frozen. Lemma 3.1. Let be the facty that causes a demand 1091
j to freeze. Then, s tentatvey open, servce g(j) s tentatvey nstaed at, andα j =max(c j,t,t g(j) ). We start by boundng the cost ncurred n openng factes, and nstang servces. Let D be the subset of demands {j : F s.t. β j > 0}. Lemma 3.2. The cost of openng factes s at most j D α j. Furthermore, the cost of nstang servces s at most j α j. Proof. For each facty that s tentatvey opened, we have that f = j D β j. By the constructon of F, we know that for each demand j, there s at most one F such that β j > 0. Summng over a factes, and usng ths fact, we see that f = β j = β j α j. F F j D j D F j D By the defnton of nbr(), we know that for any servce type, fnbr() f. For each F, we nsta servce ether at F or at nbr(), and so we can upper bound the tota cost of nstang servces of type by F f. Snce each servce s tentatvey nstaed ony when f = j G θ j,wehavethat F f = j G F θ j. The noton of servce--dependence nsures that for each demand j G,theresatmost one facty F for whch θ j s postve. We obtan that the tota cost of nstang servce s at most j G α j, whch mmedatey mpes the emma. We next bound the assgnment cost ncurred by the souton computed. The foowng facts, whch foow drecty from the constructon of the agorthm, w be usefu n ths anayss. Fact 3.1. Suppose that β k s postve. Then t foows that c k α k β k and α k t. Fact 3.2. Suppose that θ k s postve. Then c k α k θ k and c k <t g(k).ifβ k =0then α k t g(k). For exampe, we use these n dervng the foowng bounds. Cam 3.1. If and are dependent factes n F, then c < 2mn(t,t ). Proof. Let k be a cent such that β k and β k are postve. Appyng Fact 3.1 for both of these, and appyng the trange nequaty, we get that c < 2α k 2mn(t,t ). Cam 3.2. Let, F be servce--dependent due to demand k G.Thenc < 2max(t,t ) and both c k and c k are ess than α k. Proof. From the dependence of and, t foows that θ k and θ k are postve. Appyng Fact 3.2 for both of these, and usng the trange nequaty, we get that c < 2max(t,t ), c k <α k and c k <α k. Lemma 3.3. If j D, then the assgnment cost ncurred for j s at most 3α j ;fj D, then the assgnment cost ncurred for j s at most 5α j. Proof. We w show that there aways exsts some open facty wth servce g(j) nstaed that s no further from j than the camed bound (and hence the cosest one, to whch j s assgned, s no further away). Consder j D wth g(j) =. Let be the unque facty n F for whch β j s postve. If F,then we have nstaed servce at, and c j α j β j. Otherwse, and are servce--dependent for some n F F wth t t. So, by Cam 3.2, c < 2max(t,t ) = 2t. Snce β j > 0, t foows that t α j β j. So by the trange nequaty, c j 3α j. Now consder a demand j / D, and agan et g(j) =. Let be the facty that caused j to freeze, and so α j = max(c j,t,t ). If F F, then servce s nstaed at, andc j α j. Suppose that F F, and et = nbr() (an open facty at whch servce s nstaed, see Fg. 1a). By Cam 3.1, c 2t = c j 3α j. Next suppose that F.Snce F,theremust exst F such that was not pcked n F because and are servce--dependent due to a cent k. If s aso n F (Fg. 1b), then servce s nstaed there. Appyng Cam 3.2, we obtan that both c k and c k are ess than α k max(t,t ) α j, and hence c j 3α j. However, f F (so / F ), then et = nbr( ); servce s nstaed at (Fg. 1c). Snce and servce-dependent, we have that t t, c 2mn(t,t ) (Cam 3.1), and c j 3α j as above, whch mpes that c j 5α j. Ths competes the proof. Theorem 3.1. The above agorthm returns a souton of cost at most 6 j α j 6 OPT. Proof. Foows from Lemma 3.2 and Lemma 3.3. 4 An LP-Roundng Agorthm We now gve an agorthm for the speca case where f = f,.e., the nstaaton cost depends ony on the servce and not on the ocaton at whch t s nstaed. Note that ths case s handed by the prmadua agorthm above. Here we adapt the roundng procedure of [6] to gve determnstc and randomzed approxmaton agorthms achevng ratos of 6 and 2.391 respectvey. Let (x, y) and (α, β, θ) be the optma soutons to (P) and (D) respectvey. We can ensure that for 1092
(a) (b) (c) j j t t k α k k α k j α j k F F, = nbr() / F,, are servce--dependent k Facty caused j to freeze, and are servce--dependent, = nbr( ) Fgure 1: Boundng the assgnment cost of j. (a), (b) Dfferent 3-hop cases, and (c) the 5-hop case. every, j and, x j =0orx j = y g(j) and y =0or y = y. We w round the fractona souton (x, y) to an nteger souton osng a factor of at most 6. Let F j = { : x j > 0}. We descrbe the agorthm brefy. A1. Frst, for every servce type, we consder the cents n G and custer the factes on whch servce s nstaed around some custer centers: pck j G wth smaest α j vaue and form a custer around j consstng of the factes n F j. We remove every cent k G (ncudng j) thats assgned (fractonay) to some facty n the custer created, and recurse on the remanng set of cents unt no cent n G s eft. Let D be the set of custer centers. A2. Let D = D. We cannot open a facty n every custer snce dfferent custers coud share the same fractona facty weght (y ) f the custer centers request dfferent servces. Say that j, k D are dependent f F j F k φ. Note that ths can ony happen f j and k request dfferent servces. We consder cents n D n order of ncreasng α j and pck a maxma ndependent subset D. For each cent j D, we open the facty n F j wth smaest f and nsta servce g(j) ont. Further for every k D D,theressomej D wth α j α k such that j and k are dependent. We pck some such j and nsta servce g(k) onthe facty opened from F j. Ca j the neghbor of k and denote t nbr(k). A3. We assgn demand j to a facty as foows: () f j D t s assgned to the facty opened from F j, () f j D D,thennbr(j) =k D and j s assgned to the facty opened from F k, and () f j / D, theressomek D g(j) D such that α k α j and j was removed from G n step A1 because a custer was created around k; weassgn j to the same facty as k. Note that ths s a feasbe assgnment of demands to factes. Lemma 4.1. The facty openng cost s at most f y. Proof. The custers correspondng to cents n D are dsjont and we open the cheapest facty n each custer the emma foows. Lemma 4.2. Thecostofnstangservcessatmost, f y. Proof. The cost of nstang a servce s ndependent of the ocaton at whch t s nstaed, and for any servce type, we nsta servce on at most one (new) ocaton per custer center n D. So the tota cost of nstang servces s at most j D f = j D f F j y, f y. Lemma 4.3. The assgnment cost of cent j s at most 5α j. Proof. If j D, we assgn j to a facty F j,and c j α j by compementary sackness. If j D D and nbr(j) =k D, we assgn j to the facty opened from F k. Snce α k α j and c jk 2α j, c j 3α j. If j / D, j s assgned to the same facty as k where k D g(j) D and j was removed from G g(j) n step A1 because a custer was created around k.soα k α j and c jk 2α j.fromabove,k s assgned to a facty wth c k 3α k,soc j 5α j. Thus we have proved the foowng theorem. Theorem 4.1. The cost of the souton returned s at most 6 OPT. 1093
4.1 Improvement usng randomzaton. We gve a roundng procedure that combnes custered randomzed roundng [6] and the fterng based technque of [10, 14]. We defne some notaton frst. Let 0 <γ<1 be a parameter that we w set ater and r = 1 γ. Sort the factes n F j by ncreasng c j. Let be the frst facty n ths orderng such that F j :c j c j x j γ. Let N j be the subset of F j consstng of a factes (ncudng ) that come before n ths orderng. Defne C j (γ) =c j and C j = c jx j. To smpfy thngs we assume that each y γ and for any j, N j y s exacty γ. If some y >γ, then we can create at most 1/γ copes of and set y γ for each of the copes so that copes y = y (settng the varabes x j,y accordngy). Smary f N j y >γ,wecan take the facty n N j that comes ast n the orderng n F j and spt t nto two copes 1 and 2 settng y 2 = N j y γ, y 1 = y y 2 (and the other varabes accordngy). We ncude ony 1 n N j thus ensurng that N j y = γ. The cost of the fractona souton remans unchanged by these transformatons and any souton to the modfed nstance gves a souton to the orgna nstance of no greater cost. R1. Ths s the same as step A1 except that we choose j G wth smaest 2α j +C j (γ)+ C j as the custer center. Ths gves us a set of custer centers D for each servce type. R2.WeprunethesetD = D as n step A2 but modfy the noton of dependency to say that j, k D are dependent f N j N k φ, and consder the cents n D n ncreasng order of C j (γ)+ C j.for k D D we defne nbr(k) as before. We ca the factes n N j for cents j D centra factes, and the rest as non-centra factes. R3. For every cent j D we randomy open exacty one facty n N j by choosng facty wth probabty y / N j y = r y. Ths facty now serves as a backup facty for a the cents that woud get assgned to ths facty n step A3 of the determnstc agorthm. R4. Independent of step R3, each non-centra facty s opened ndependenty wth probabty r y. R5. For any facty, be t a centra or a non-centra facty, f s opened (n R3 or R4), we nsta on t a servces that are nstaed on t n the fractona souton,.e., a such that y > 0. R6. For every cent j D D, f no facty from F j s open, we nsta servce g(j) on the facty opened n R3 from N nbr(j). R7. We assgn demand j to the nearest open facty at whch servce g(j) s nstaed. Lemmas 4.1 and 4.2 are modfed to the foowng. Lemma 4.4. The expected cost of openng factes s r f y. The expected cost of nstang servces s at most ( ) r + 1 e, f r y. Proof. Each facty s opened wth probabty r y. The cost of nstang servces n step R5 s bounded by Pr[ s opened (n R3 or R4)] :y >0 f = r y :y >0 f = r, f y snce y > 0 = y = y. Consder cent j D D wth g(j) =. For every non-centra facty F j,ete be the event that s opened n step R4 and p =Pr[E ]=r y. For every custer center k D such that S k = F j N k φ, et E k be the event that a facty from S k s open after step R3. Let p k =Pr[E k ]= P S y k N y = r S k y. k Let m be the tota number of events. A the events E are ndependent. The probabty that servce s nstaed n step R6 due to cent j, s the probabty that no facty from F j s open after steps R3 and R4, whch s at most m n=1 (1 p n) e P n pn = e r. So the cost of nstang servces n step R6 s at most 1 e r j D D f g(j) 1 e, f r y snce F j y =1and P any two cents n D have dsjont F j. To bound the assgnment cost, we bound the assgnment cost ncurred under a provaby worse way of assgnng demands to factes. Demand j s assgned to a facty as foows. If some facty F j s open, we assgn demand j to the nearest such facty. Otherwse f j D D, j s assgned to ts backup facty. If j / D, there s some cent k D g(j) D such that j was removed from G g(j) because a custer was formed around k n step R1. We assgn j to the same facty as k; soj may be assgned ether to a facty n F k or to ts backup facty n N nbr(k),fk/ D and no facty from F k s open. Note that servce g(j) snstaedonthe facty to whch j s assgned. We need the foowng emma from [6] (see aso [15]). Lemma 4.5. Let d 1 d 2... d m and 0 p n 1 for n =1,...,m.Then, p 1 d 1 +(1 p 1 )p 2 d 2 + +(1 p 1 ) (1 p m 1 )p m d m n m p nd n (1 n m p ) (1 p n ). n n m Lemma 4.6. Let j be any demand. Let X be the dstance between j and the facty assgned to t and Z be the event that no facty F j s open. Then, 1094
() If j / D, E [ X Z ] 3α j + C j (γ) + C j, () E [ X ] C j + 1 e r (3α j + C j (γ)). Proof. If j D,E [ X ] = N j c j x j / N j x j C j snce every facty n F j N j s farther from j than every facty n N j.forj/ D, we show () and use t to prove (). Suppose j D D, k = nbr(j) anda = N j N k φ. For any facty A we have c j α j and c k C k (γ). Let B be the dstance between j and ts backup facty n N k. Event Z mpes that j s assgned to the backup facty n N k so condtoned on Z, X = B. Iftheressome A such that c k C k we have a determnstc bound of B α j + C k + C k (γ). If there s no such n A, snce the uncondtona dstance between k and the backup facty n N k s at most C k, by condtonng on Z we are ony removng weght from factes that are farther than the average dstance. So the condtona expected dstance between k and the backup facty s at most C k mpyng that E [ B Z ] =E [ X Z ] α j + C k (γ)+ C k. In ether case E [ X Z ] α j + C j (γ) + C j, where the ast nequaty foows snce we ook at cents n D n ncreasng order of C j (γ)+ C j and k was pcked before j. If j/ D, there must be a cent k D g(j) such that j was removed from G g(j) because a custer was formed around k n step R1. So j, k are assgned to the same facty, 2α k + C k (γ) + C k 2α j + C j (γ) + C j and c jk α j + α k. If a facty n F k s open then we have a determnstc bound of X α j +2α k. Otherwse j, k are assgned to the backup facty for k and by the above bound on E [ X Z ] for k, the condtona expected dstance from j to the backup facty s at most α j +2α k + C k (γ)+ C k 3α j + C j (γ)+ C j. We now prove part (). For a non-centra facty F j,etp and E be as defned n Lemma 4.4 and et d = c j. For every k D such that S k = F j N k φ, et p k,e k be as defned n Lemma 4.4 and defne d k = E [ ] dstance from j to S k E k = S k c j y / S k y. Let the dstances be ordered so that d 1 d 2... d m where there are m events n a. Snce the events E are ndependent and y = x j, p =Pr[Z] = m n=1 (1 p n) e P n pn = e r.so, E [ X ] p 1 d 1 +(1 p 1 )p 2 d 2 + +(1 p 1 ) (1 p m 1 )p m d m + p E [ X Z ] n m p nd n n m p (1 p)+p(3α j + C j (γ)+ C j ) n C j + 1 e r (3α j + C j (γ)). Theorem 4.2. The randomzed agorthm produces a souton of expected cost at most, ( max r + 4 e r, 1+ 1 (1 γ)e r + 3 ) e r OPT where r =1/γ. Takngγ =0.67674 we get a souton of cost at most 2.391 OPT. Proof. From the defnton of C j (γ) and the Markov property we have, C j (γ) C j 1 γ. The proof now foows from Lemmas 4.4 and 4.6. 5 The k-medan Varant We consder a varant of ths probem where we have the addtona constrant that at most k factes may be opened. Ths adds the constrant y k to the near program (P). The objectve functon of the dua (D) gets modfed to max j α j kz and constrant (2.2) changes to j β j f + z. Let (KP) and (KD) be the modfed prma and dua programs and OPT K be the vaue of an optma k-medan souton. 5.1 A modfed prma-dua agorthm. We frst modfy the prma-dua agorthm of Secton 3 to obtan the stronger guarantee that we return a souton to (P) of cost (O, I, C), and a souton (α, β, θ) to(d)such that 6O + I + C 6 j α j 6 OPT,whereO, I, C denote respectvey the facty openng cost, the servce nstaaton cost and the cent assgnment cost. We descrbe the agorthm brefy and sketch the anayss. The dua ascent process s the same as n Secton 3 but we modfy the way n whch we open factes and nsta servces to ensure that a demand j does not pay for both openng a facty and for nstang a servce at some other facty. To do ths we consder a more detaed noton of dependence between factes. We cassfy 4 types of dependence between factes. ordered par (, )s, Say that the (1) ff-dependent (f for facty) f there s a demand j such that β j,β j > 0. (2) sf- dependent (s for servce) f there exsts j G such that θ j,β j > 0. (3) ss- dependent f for some j G,bothθ j,θ j > 0. (4) fs- dependent f for some j G,bothβ j,θ j > 0. Reca that F s the set of tentatvey open factes, F F the set of tentatvey open factes on whch servce s tentatvey nstaed, t s the tme at whch facty F became tentatvey open, and for F, t s the tme at whch servce was tentatvey nstaed at. Intay for each facty F,etS be the set of servces that are tentatvey nstaed at. 1095
M1. We frst pck a set F F of factes to open, and for each F asett S of servces to nsta at facty. Intay F = φ and T = φ for a. We consder factes n F n the order gven by O. Whe F φ, 1. Let F be the currenty consdered facty. Remove from F,setF F {}, T = S. 2. For each F we do the foowng. a) If (, ) s ff-dependent or T s.t. (, )ssf- dependent or S s.t. (, )sfs- dependent and t <t,set F F { }. Ca the neghbor of and denote t by nbr( ). Otherwse, b) For every S,f(, )sfs- dependent (so t t ) or T and (, ) s ss- dependent, set S S {}. We open the factes n F and for each F nsta a of the servces n T at. M2. We now nsta servces at some more factes. Consder servce type. Let A be the factes n F at whch servce s nstaed (.e., T ). Note that A F F. Let B = F F. We remove from B every facty for whch there s some facty F such that (1) (, )sfs- dependent, or (2) A and (, ) s ss- dependent. We say that a set of factes s ss- ndependent f no par (, ) of factes from the set s ss- dependent. We pck a maxma ss- ndependent subset F B. Intay F = φ. We consder factes n B n ncreasng order of t and add facty to F f F {} remans ss- ndependent. For every F we nsta servce on facty nbr() F. M3. Each cent j s assgned to the nearest open facty at whch servce g(j) s nstaed. Anayss Sketch. Consder the set of demands D = {j : F s.t. β j > 0}. By the constructon of F,we know that for each demand j theresatmostone F such that β j > 0; for j D et o(j) denote ths unque facty n F. By desgn we ensure that any demand j G contrbutes θ j > 0 for at most one facty A F and further that f j D then = o(j) s the ony such facty. Ths gves the foowng. Lemma 5.1. The cost of openng factes s j D β o(j)j. The cost of nstang servces s at most j D θ o(j)j + j/ D α j. The bound on the assgnment cost ncurred s smar to the bound n Lemma 4.3 and s proved smary. Lemma 5.2. If j D, the assgnment cost of j s at most 3(α j β o(j)j ). If j / D the assgnment cost ncurred for j s at most 5α j. Combnng the above two emmas and the fact that for j D, α j = c o(j)j + β o(j)j + θ o(j)j, we get the foowng theorem. Theorem 5.1. The souton returned has cost (O, I, C) such that 6O + I + C 6 j α j 6 OPT. 5.2 An 18-approxmaton agorthm. Suppose we fx z and run the above prma-dua agorthm wth the facty costs modfed to f + z. Suppose the agorthm returns a prma souton of cost (O, I, C) thatopens k factes and a dua souton (α, β, θ). Here O s the facty openng cost wth the orgna costs f. Then, we can show that we have a souton of cost at most 6 OPT K.snce(α, β, θ, z) s a feasbe souton to (KD) and by Theorem 5.1, 6(O+kz)+I +C+ 6 j α j = 6O+I+C 6( j α j kz) 6 OPT K. So our goa s to fnd such a z n poynoma tme. We do not qute know how to do ths, nstead we fnd two vaues z 1 >z 2,cose together such that the agorthm opens k 1 <kfactes for z 1 and k 2 > k factes for z 2. When z s very arge, e.g., D (max j c j +max f ), the agorthm w open just one facty and at z = 0 the agorthm opens >k(we assume ths otherwse the souton at z =0 costs at most 6 OPT K snce (α, β, θ, 0) s a feasbe dua souton). We perform a bsecton search n ths range to fnd z 1 and z 2. We combne these two soutons to frst get a fractona souton of cost 6(1 ɛ) 1 OPT K,for some sma ɛ, that (fractonay) opens k factes and n whch each demand s assgned to at most two factes. If the servce cost depends ony on the servce type, we can round ths souton osng a factor of 3(1 ɛ) to get an nteger souton that opens exacty k factes, thus gettng an overa approxmaton rato of 18. Ths deawasusedn[9]forthek-medan varant of UFL. However our fna roundng procedure dffers from the one n [9] snce we aso have to pay for nstang servces and need to ensure that demand j s connected to an open facty at whch servce g(j) s nstaed. Theorem 5.2. If the servce nstaaton cost depends ony on the servce and not on the ocaton, we can get an 18-approxmaton agorthm for the k-medan varant of facty ocaton wth servce nstaaton costs. 6 Extensons Arbtrary Demands. Our resuts carry over to the case where nstead of unt demands, cent j may have a demand d j 0. We can reduce ths to the unt demand case by makng d j copes of cent j, but ths makes 1096
the agorthm run n pseudo-poynoma tme. We can smuate ths reducton however. In the prma-dua agorthm we rase α j at rate d j and say that j has reached f α j d j c j. In the LP roundng agorthms of Secton 4 the ony change s that α j gets repaced wth α j /d j n steps A1, A2 and R1. The anayss n Sectons 3, 4 and 5 extends n a straghtforward way and we get the same approxmaton guarantees. [14] D. B. Shmoys, É. Tardos, and K. I. Aarda. Approxmaton agorthms for facty ocaton probems. In Proceedngs of the 29th Annua ACM Symposum on Theory of Computng, pages 265 274, 1997. [15] M. Svrdenko. An mproved approxmaton agorthm for the metrc uncapactated facty ocaton probem. In Proceedngs of 9th IPCO, pages 240 257, 2002. [16] C. Swamy. A note on the data pacement probem. Unpubshed manuscrpt, 2003. Acknowedgments. We thank Robn Roundy for stmuatng dscussons and hepfu suggestons. References [1] N. Aksoy and S. Erenguc. Mut-tem nventory modes wth coordnated repenshments: a survey. Internatona Journa of Operatons & Producton Management, 8:63 73, 1988. [2] A. Archer, R. Rajagopaan and D. Shmoys. Lagrangan reaxaton for the k-medan probem: new nsghts and contnuty propertes. In Proceedngs of 11th ESA, pages 31 42, 2003 [3] I. Baev and R. Rajaraman. Approxmaton agorthms for data pacement n arbtrary networks. In Proceedngs of the 12th Annua ACM-SIAM Symposum on Dscrete Agorthms, pages 661 670, 2001. [4] Y. Barta, M. Charkar and D. Raz. Approxmatng mn-sum k-custerng n metrc spaces. In Proceedngs of the 33rd Annua ACM Symposum on Theory of Computng, pages 11 20, 2001. [5] F. A. Chudak. Improved approxmaton agorthms for uncapactated facty ocaton. In Proceedngs of 5th IPCO, pages 180 194, 1998. [6] F. Chudak and D. Shmoys. Improved approxmaton agorthms for the uncapactated facty ocaton probem. SIAM Journa on Computng. Toappear. [7] U. Fege. A threshod of n n for approxmatng set cover. Journa of the ACM, 45:634 652, 1998. [8] K. Jan, M. Mahdan, E. Markaks, A. Saber, and V. Vazran. Greedy facty ocaton agorthms anayzed usng dua-fttng wth factor-reveang LP. Journa of the ACM. Toappear. [9] K. Jan and V.V. Vazran. Approxmaton agorthms for metrc facty ocaton and k-medan probems usng the prma-dua schema and Lagrangan reaxaton. Journa of the ACM, 48:274 296, 2001. [10] J. H. Ln and J. S. Vtter. ɛ-approxmatons wth mnmum packng constrant voaton. In Proceedngs of the 24th Annua ACM Symposum on Theory of Computng, pages 771 782, 1992. [11] M. Mahdan, Y. Ye, and J. Zhang. Improved approxmaton agorthms for metrc facty ocaton. In Proceedngs of 5th APPROX, pages 229 242, 2002. [12] P. Mrchandan and R. Francs, eds. Dscrete Locaton Theory. John Wey and Sons, Inc., New York, 1990. [13] D. B. Shmoys. Approxmaton agorthms for facty ocaton probems. In Proceedngs of 3rd APPROX, pages 27 33, 2000. 1097