Approximating the Two-Level Facility Location Problem Via a Quasi-Greedy Approach. Jiawei Zhang. October 03, 2003

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1 Approximating th Two-Lvl Facility Location Problm Via a Quasi-Grdy Approach Jiawi Zhang Octobr 03, 2003 Abstract W propos a quasi-grdy algorithm for approximating th classical uncapacitatd 2-lvl facility location problm 2-LFLP). Our algorithm, unlik th standard grdy algorithm, slcts a sub-optimal candidat at ach stp. It also rlats th minimization 2-LFLP problm, in an intrsting way, to th maximization vrsion of th singl lvl facility location problm. Anothr fatur of our algorithm is that it combins th tchniqu of randomizd rounding with that of dual fitting. This nw approach nabls us to approximat th mtric 2-LFLP in polynomial tim with a ratio of 1.77, a significant improvmnt on th prviously known approximation ratios. Morovr, our approach rsults in a local improvmnt procdur for th 2-LFLP, which is usful in improving th approximation guarants for svral othr multi-lvl facility location problms. An additional rsult of our approach is an Olnn))-approximation algorithm for th non-mtric 2-LFLP, whr n is th numbr of clints. This is th first non-trivial approximation for a non-mtric multi-lvl facility location problm. Kywords. Two-lvl facility location, approximation algorithm, linar programming rlaxation, quasi-grdy approach. An xtndd abstract of this papr is to appar in th Procdings of th 15th ACM-SIAM Symposium on Discrt Algorithms SODA), January Dpartmnt of Managmnt Scinc and Enginring, Stanford Univrsity, Stanford, CA, jiazhang@stanford.du. Mailing Addrss: P.O.Box 11148, Stanford, CA,

2 1 Introduction 1.1 Problm statmnt In th singl lvl uncapacitatd facility location problm, w ar givn a st of clints and a st of facilitis. W want to opn a subst of th facilitis such that all th clints ar srvd by th opn facilitis and th total cost of opning facilitis and srving clints is minimizd. In th k-lvl uncapacitatd facility location problm k-lflp), th dmands must b routd among facilitis in a hirarchical ordr, i.., from th highst lvl th factoris) down to th lowst th rtailrs), bfor raching th clints. Th k-lflp ariss naturally in dsigning logistic systms. Th k-lflp can b formulatd formally as follows. W ar givn a st of clints D and k lvl sts of facilitis F 1, F 2,, F k. Dnot P = F 1 F 2 F k and F = k t=1 F t. Each clint j D must b srvd by an opn path p = i 1, i 2,, i k ) P of k facilitis with xactly on from ach of th k lvls, whr a path p is opn if and only if vry facility on th path is opn. Thr is a facility cost f it for opning facility i t F t 1 t k). Furthrmor, If clint j D is srvd by an opn path p = i 1, i 2,, i k ) P a connction cost c jp is incurrd whr c jp = c ji1 + k t=2 c i t 1 i t and c ji is th connction cost btwn j and i for j, i D F. Hr, w wish to opn a subst of facilitis such that ach clint is assignd to an opn path and th total cost is minimizd, i.., to choos S t F t, t = 1, 2,, k, such that min c jp + p S 1 S 2 S k j D k f it t=1 i t S t is minimizd. W also assum that th connction costs ar nonngativ, symmtric, and satisfy th triangl inquality, i.., for ach i, j, l D F, c ij 0, c ij = c ji and c ij c il + c lj. In this papr w ar concrnd with th 2-LFLP, th most studid spcial cas of k-lflp in th litratur of Oprations Rsarch for k 2 [1, 8, 17, 18, 21, 29, 30, 31, 34]. Th study on th 2-LFLP is motivatd by th fact that in many applications, spcially in supply chains, thr ar hirarchical two-lvl structurs. Th problm also has applications in tlcommunications and computr ntwork dsigns [8]. On th othr hand, although it is th simplst modl among all th k-lflp for k 2, th 2-LFLP has som fundamntal structural diffrncs from th 1-LFLP. For xampl, th 2-LFLP dos not possss th so-calld suprmodularity, a wll-known proprty for th 1-LFLP [22]. This proprty is oftn hlpful in dsigning branch-and-cut algorithms and in analyzing som approximation algorithms. Thus, th 2-LFLP nds nw tchniqus. 1.2 Prvious rsults Th 2-LFLP gnralizs th popular singl lvl uncapacitatd facility location problm 1-LFLP), which is alrady NP-hard [13]. Sinc th work of Shmoys, Tardos and Aardal [33], dsigning approximation algorithms for th 1-LFLP and its rlatd problms has rcivd considrabl attntion during th past fw yars [32]. W call an algorithm of a minimization problm a ρ 1)- approximation algorithm if for any instanc of th problm th algorithm runs in polynomial tim 2

3 and outputs a solution that has a cost at most ρ tims th minimal cost, whr ρ is calld th prformanc guarant or th approximation ratio of th algorithm. Guha and Khullr [19] provd that th xistnc of a polynomial tim approximation algorithm for th 1-LFLP would imply that P = NP. And th bst currntly known approximation ratio for th 1-LFLP is 1.52 du to Mahdian, Y and Zhang [26]. Thrfor, th approximability of th 1-LFLP is wll undrstood. Howvr, th 2-LFLP rmains intriguing. On can asily s that th lowr bound of th 1-LFLP also applis to th 2-LFLP, and no bttr lowr bound is known. In [33], th algorithm for th 1-LFLP has bn xtndd to th 2-LFLP with an approximation ratio Latr on, Aardal, Chudak and Shmoys [2] showd that th k-lflp can b approximatd in polynomial tim by a factor of 3 for any k 2 using a linar programming rlaxation. Howvr, thir algorithm dos not possss a bttr prformanc guarant for k = 2, nithr do a sris of rcntly proposd fastr combinatorial algorithms du to Myrson, Munagala, and Plotkin [28], Guha, Myrson, and Munagala [24], Bumb and Krn [7], and Agv [3]. In fact, th algorithms of [2, 7, 3] will produc solutions whos opn paths ar disjoint, and Edwards [15] showd that such algorithms can not hav worst cas ratios that ar bttr than 3 vn for k = 2. Vry rcntly, Agv, Y, and Zhang [5] proposd two diffrnt combinatorial algorithms and showd that th bttr of th two has a prformanc guarant 2.43, although ach of thm has a prformanc guarant at last Our rsults and tchniqus Th main rsult of this papr is that th 2-LFLP can b approximatd in polynomial tim by a factor of 1.77, a significant improvmnt ovr th prvious prformanc guarants, sinc no on can do bttr than unlss P = NP. Th improvd ratio is achivd by using what w call a quasi-grdy approach. Our algorithm is analyzd by using th tchniqu of factor-rvaling LP dvlopd by Jain, Mahdian, and Sabri [25]. On advantag of our algorithm is that it can b asily gnralizd to solv th so-calld two lvl concntrator location problm s [9] and th rfrncs thrin) with th sam ratio Th quasi-grdy approach also rsults in a local improvmnt procdur for th 2-LFLP, which dos not improv th ratio 1.77 for th 2- LFLP but is usful in improving th prformanc guarants for othr multi-lvl facility location problms. In particular, w show that 3- and th 4-LFLP can b approximatd by factors of 2.51 and 2.81, rspctivly, obtaining th currnt bst approximation ratios for ths two problms. W also obtain an improvd 3-approximation algorithm for th 2-LFLP with soft capacitis. An additional rsult of our approach is an Oln D ))-approximation algorithm for th non-mtric 2-LFLP whr th connction cost may not satisfy th triangl inquality. This is th first nontrivial approximation algorithm for a non-mtric multi-lvl facility location problm. And its approximation ratio matchs th bst known approximation ratio for th non-mtric 1-LFLP du to Hochbaum [20]. Furthrmor, th approximation ratio is th bst possibl up to a constant, unlss P=NP [16]. Grdy algorithms hav bn succssful in tackling th 1-LFLP [25, 26]. In a standard grdy approach, at ach stp, on computs a grdy function valu for ach lmnt of a candidat st and chooss th optimal candidat basd on ths valus. Whn applid to th 1-LFLP, th candidat 3

4 st is th st of unopn) facilitis, and th grdy function is th ratio of th cost incurrd to th numbr of nw clints srvd, which can b computd asily. Howvr, th issu is mor complicatd whn w apply th grdy approach to th 2-LFLP. In particular, it is difficult to dfin a candidat st. Som straightforward choics do not work such as F 1 F 2 and F 1 F 2. On sophisticatd dfinition works whr a candidat is dfind as many facilitis with xactly on in F 2 and th rst in F 1. Unfortunatly, th problm is that thr ar xponntially many candidats, and w can not choos th bst candidat by comparing thir grdy function valus with ach othr in polynomial tim. Thus, w mak a simpl but important obsrvation for th 2-LFLP: choosing th bst candidat among th xponntially many candidats according to an asy grdy function is quivalnt to choosing th bst candidat among polynomially many candidats according to an appropriatly dfind hard grdy function. In th lattr, givn a candidat, it is NP-hard to comput th xact valu for th hard grdy function. But th good nws is that w may comput th grdy function valu approximatly in polynomial tim. Thrfor, w could choos th bst candidat according to th approximatd grdy function valu. W call this approach quasi-grdy sinc it may not choos th grdist candidat. It turns out that computing th hard grdy function valu for solving th 2-LFLP is quivalnt to th so-calld maximization vrsion of th 1-LFLP Max-1-LFLP in short). Rcall that in th minimization 1-LFLP, assigning a clint to an opn facility will incur a connction cost. In th Max-1-LFLP, a rvnu will b gnratd by assigning a clint to an opn facility, and th objctiv is to maximiz th nt profit th total rvnu minus th total facility cost). Th Max-1-LFLP and th minimization 1-LFLP ar quivalnt from th prspctiv of optimization, but not from that of approximation. Approximation algorithms for th Max-1-LFLP hav a longr history than thos for th 1-LFLP[12, 4]. Howvr, th rsults of [12, 4] do not hlp in stablishing our rsult for th 2-LFLP. W shall giv a nw approximation algorithm for th Max-1-LFLP such that th approximation rsult can b usd in proving our bound for th 2-LFLP. To th bst of our knowldg, this is th first tim that approximations for th maximization and th minimization vrsions of a facility location problm hav bn rlatd. W rmark that th quasi-grdy approach has bn usd in dsigning approximation algorithms in othr sttings, for xampl, Chkuri, Evn and Kortsarz [11]. Howvr, our approach is nw in that it rducs th siz of th st of candidats from xponntially many to polynomially many in such a way that th grdy function for th lattr can b wll approximatd. Our algorithm and analysis combin th tchniqu of randomizd rounding with that of dual fitting. Both tchniqus hav bn usd for solving various facility location problms, but nvr combind bfor. In our algorithm, ach grdy stp is solvd by randomly rounding th solution of its linar programming rlaxation. Dual fitting and th factor-rvaling LP) is usd for proving th ovrall prformanc guarant. 4

5 1.4 Organization of th papr Th rst of th papr is organizd as follows. In Sction 2, w prsnt a linar programming basd approximation algorithm for th Max-1-LFLP. Th quasi-grdy algorithm for th 2-LFLP and its analysis is givn in Sction 3. In Sction 4, a local improvmnt procdur basd on th quasigrdy approach for th 2-LFLP is proposd. In sction 5, w prsnt improvd approximation algorithms for th non-mtric 2-LFLP, th two lvl concntrator location problm, and othr multi-lvl facility location problms. Final rmarks ar givn in Sction 6. 2 An algorithm for th Max-1-LFLP In this sction, w considr th maximization vrsion of th singl lvl) facility location problm Max-1-LFLP). In th Max-1-LFLP, w ar givn th st of clint D, th st of facilitis F. Th facility cost for opning facility i is f i and, Th rvnu gnratd by assigning clint j to facility i is d ij 0. Hr d ij may not satisfy th triangl inquality. Th objctiv is to opn a subst of th facilitis of F and thn assign ach of th clints in D to an opn facility such that th nt profit is maximizd. Th Max-1-LFLP can b formulatd as th following intgr program. Max f i y i 1) s.t. d ij x ij i F,j D i F x ij = 1 for all j D i F x ij y i for all i F, j D x ij, y i {0, 1} for all i F, j D whr y i = 1 if w dcid to opn facility i, othrwis y i = 0; x ij = 1 if clint j is assignd to facility i, othrwis x ij = 0. Considr any fasibl solution, whos total profit is assumd to b C F, whr C and F corrspond to th total rvnu and th total facility cost, rspctivly. Our goal is to find an algorithm that producs a solution with profit C F 1 1 )C F. Th algorithm is basd on th linar programming LP) rlaxation of this problm. Howvr, th actual LP w will solv is diffrnt from th xact LP rlaxation of 1). Algorithm MAX Stp 1. Solv th following LP and obtain an optimal solution x, y): Max 1 1 ) s.t. d ij x ij i F,j D i F x ij = 1 for all j D i F x ij y i for all i F, j D 0 x ij, y i 1 for all i F, j D 5 f i y i 2)

6 Stp 2. For ach i F, opn facility i indpndntly with probability y i, and assign ach clint j D to an opn facility with maximum rvnu. Lt th rsultd solution b ˆx, ŷ) and th corrsponding facility cost and rvnu b F and C rspctivly. W first prov a tchnical lmma. Lmma 1. Givn 3n rals d i, t i and p i, i = 1, 2,, n such that d 1 d 2 d n 0, 0 t i p i 1 and n t i = 1, thn ) n k 1 n n d k 1 p i )p k 1 1 t i ) d i t i. k=1 Proof. Lt On can vrify that g z k = = gz 1, z 2,, z n ) = 0 n k=1 k 1 1 z i ) d k k 1 1 z i )d k 1 k 1 1 z i )d k n j=k+1 1 z i )z k. d k k 1 n j=k+1 n j=k+1 d j z j z j 1 z j ) j 1 l=k+1 j 1 l=k+1 1 z l ) 1 z l ) Thrfor, gz 1, z 2,, z n ) is nondcrasing function of z i for ach i = 1, 2,, n. It follows that gp 1, p 2,, p n ) gt 1, t 2,, t n ). Lt D = d And lt c i = D d i for i = 1, 2,, n. Thrfor, 0 c 1 c 2 c n. Furthrmor, by assumption, n t i = 1, thn by Lmma 10 of Chudak and Shmoys [14], w must hav ) n k 1 n n c k 1 t i )t k 1 1 t i ) c i t i. It implis that k=1 n k 1 D d k ) 1 t i )t k 1 k=1 k=1 Rarrang th abov inquality, w gt n 1 t i )t k 1 d k k 1 6 ) n 1 t i ) ) n 1 t i ) n D d i )t i. n d i t i.

7 Thrfor, gp 1, p 2,, p n ) gt 1, t 2,, t n ) 1 which complts th proof. ) n n 1 t i ) d i t i, Now w ar rady to prsnt th main rsult of this sction. Thorm 1. Thr xists an algorithm that finds a solution with profit C F such that C F 1 1 )C F. Proof. By th dfinition of ˆx, ŷ), ŷ i is 1 with probability y i, and 0 with probability 1 y i. Thn E[ŷ i ] = y i and thus th xpctd facility cost is E[F ] = E[ i F f i ŷ i ] = i F f i E[ŷ i ] = i F f i y i. It is lft to analyz th quantity E[C] = E[ i F,j D d ij ˆx ij ]. Considr any j D, and assum that th facilitis ar indxd such that d ij d i+1)j. For simplicity, w drop th subscript j and dnot th rvnus by d 1, d 2,, such that d i d i+1. By conditional probability, E[ d iˆx ij ] = k 1 ) d k 1 y i ) y k. i F 1 k F Furthrmor, 0 x kj y k and k F x kj = 1 for vry j. Thrfor, by Lmma 1, th xpctd rvnu that j can gt is boundd blow by 1 i F1 ) x ij ) d i x ij. i F d i x ij 1 1 ) i F Th last inquality holds sinc i F 1 x ij ) 1 x ij ) F 1 1 F F ) F 1. Thrfor, E[ i F i F,j D d ij ˆx ij ] E[ i F f i ŷ i ] 1 1 ) i F,j D d ij x ij i F Sinc x, y) is th optimal solution for th LP, th right hand sid of th abov inquality is boundd blow by 1 1 )C F. Thrfor, E[C F ] 1 1 )C F. f i y i. 7

8 By th standard tchniqu using conditional xpctation, w can drandomiz th algorithm such that C F 1 1 )C F. 3 Th quasi-grdy algorithm for th 2-LFLP For our purpos, it would b usful to hav th following dfinition. Dfinition 1. An algorithm is calld a R f, R c )-approximation algorithm for th k-lflp, if for vry instanc I of th k-lflp, and for vry solution SOL for I with facility cost F SOL and connction cost C SOL, th cost of th solution found by th algorithm is at most R f F SOL +R c C SOL. W cit a lmma from Mahdian t al [26]. Lmma 2. [26] For a, b) = 1.104, 1.78) or a, b) = 1.118, 1.77), k α i a k m i + b f, if th following systm of inqualitis holds 1 j < k : α j α j+1 1 l < j < k : r l,j r l,j+1 1 l < j k : α j r l,j + m j + m l 1 j k : j 1 l=1 maxr l,j m l, 0) + k l=j maxα j m l, 0) f 1 l j k : α j, m j, f, r l,j 0. 3) Furthrmor, th 1-LFLP can b approximatd by a factor of a+lnδ), 1+b 1)/δ) for any δ 1. Blow ar th dtails of our quasi-grdy algorithm for th 2-LFLP. Th algorithm is prsntd in a way similar to that of Jain t al [25]. Algorithm QG 1. W introduc a notion of tim. Th algorithm starts at tim 0. At this tim, all clints ar unconnctd and all facilitis ar unopn. Lt U b th st of unconnctd clints. Thus at this tim U = D. And α j = 0 for ach j D. At ach momnt, vry clint j will hav som mony B j availabl to offr to ach unopn facility in F 2, whr B j = α j if j is unconnctd, and B j = c jp if j is currntly connctd to an opn path p. Th amount of offrs rcivd by a facility i F 2 is computd as follows. Considr any i F 2. For ach j D and k F 1, dfin d kj = max{b j c jki, 0} whr c jki = c jk + c ki, and ˆf k = 0 if k F 1 is alrady opn; ˆfk = f k othrwis. Thn w obtain an instanc of th Max-1-LFLP. W solv this instanc by Algorithm MAX and obtain a fasibl solution. This profit could b ngativ) is th amount of offrs rcivd by th facility i. Not that ach clint j can mak an offr to a facility i F 2 through xactly on facility σ i j) F 1 whr σ i j) is th facility to which j is assignd by Algorithm MAX. Th contribution mad by clint j to facility i is qual to d σi j)j. 8

9 2. Whil U, incras th tim and simultanously incras α j at th sam rat for ach j U, until on of th following vnts occurs: 1). For a unopn facility i F 2, th total amount of offrs that it has rcivd from th clints is qual to f i. In this cas, w opn facility i. And for ach clint j D who has mad non-zro contribution to i, w opn facility σ i j) F 1 and assign j to th path σ i j), i). Furthrmor, if j U, thn rmov j from U and stop incrasing α j ). 2). For a clint j U and opn facilitis i F 2 and k F 1, α j = c jki, thn assign j to th path k, i) and rmov j from U and stop incrasing α j ). Rmark 1. For ach clint j D, th valu α j will incras until j gts connctd to an opn path, and α j will not chang aftr that. At ach momnt, th valu B j is usd to dnot th amount of mony availabl to B j to offr. Bfor j is connctd, B j = α j. Th first tim j is connctd to an opn path, say p, B j = c jp. It is clar that B j c jp. Aftr j is connctd, B j is th connction cost paid by clint j. For xampl, clint j may gt connctd latr to anothr opn path p with c jp c jp to sav th connction cost. Thn as long as j is connctd to p, B j = c jp. Thrfor, th valu B j will not stop incrasing until j gts connctd to an opn path it may start dcrasing from thn on). Rmark 2. In ordr to implmnt Algorithm QG in polynomial tim, w notic that th total numbr of possibl vnts is boundd by D + F 2. At any tim, w nd to find th minimum valu of how much th α j s should incras such that th nxt vnt will occur. This can b don in polynomial tim but not strongly polynomial tim) by prforming a bisction sarch. Anothr way for implmnting th algorithm is that w discrtiz th tim and only considr th valus of α j s that ar powrs of 1+ɛ), i.., {0, 1, 1+ɛ), 1+ɛ) 2, 1+ɛ) 3,, } for any givn constant ɛ > 0. Thrfor, th algorithm can b implmntd in polynomial tim for any givn constant ɛ > 0. Now, w ar rady to analyz th algorithm. First, w hav Lmma 3. Th total cost of th solution producd by Algorithm QG is j D α j. W can assum that th opn paths of an optimal solution of th 2-LFLP form a forst [15]. In ordr to analyz th prformanc guarant, w considr any tr of th forst. Th root of this tr must b a facility i F 2. And th lavs of th tr ar a subst of th facilitis, say S i F 1. W also considr th st of clints, dnotd by D i, who ar assignd to th tr rootd at i in th optimal solution. Thrfor, th total cost of th optimal solution) associatd with this tr is f i + f k + min c jki. k S i k S i j D i If w could prov that, for ach i F 2, α j R f f i + f k + R c min c jki, k S i j D i k Si j D i thn Algorithm QG must b a R f, R c )-approximation algorithm for th 2-LFLP. Thus, in th rst of 9

10 this sction, w considr a particular i, and th associatd S i and D i. W assum that D i = n and lt m j = min k Si c jki. Furthrmor, without losing gnrality, w assum that α 1 α 2 α n. For ach j : 1 j n, considr th situation of th algorithm at tim t = α j. For ach l j 1, if l is connctd to a path p bfor tim t i.. l was connctd to a path at tim α j /1 + ɛ), thn lt r l,j = c lp ; othrwis, lt r l,j = α l. In th lattr cas, α l = α j. Lmma 4. Lt f = fi + ) k S i f k, thn th systm of inqualitis 4) holds: 1 1 j < k : α j α j+1 1 l < j < k : r l,j r l,j+1 1 l < j k : α j /1 + ɛ) r l,j + m j + m l 1 j k : j 1 l=1 maxr l,j/1 + ɛ) m l, 0) + k l=j maxα j/1 + ɛ) m l, 0) f 1 l j k : α j, m j, f, r l,j 0. 4) Proof. First of all, th inquality α j α j+1 holds by assumption. And for ach l : 1 l < j n, w hav r l,j r l,j+1 sinc onc a clint is connctd to a path, it will nvr b connctd to a path with a highr connction cost. Considr clints j > l at tim t = α j. If l is unconnctd at tim t/1 + ɛ), thn by dfinition, r l,j = α l and it must b th cas α j = α l. Thrfor, α j r l,j + m j + m l. Now w assum that l is connctd to a path p at tim t/1 + ɛ). It follows that r l,j = c lp and p must b opn at tim t/1 + ɛ). Thus t/1 + ɛ) c jp, othrwis j could b connctd to p bfor tim t. Howvr, by triangl inquality, c jp c lp + m j + m l. Thus, w also hav α j /1 + ɛ) = t/1 + ɛ) r l,j + m j + m l. It is lft to prov that for all j, j 1 max r l,j /1 + ɛ) m l, 0) + l=1 n max α j /1 + ɛ) m l, 0) l=j 1 f i + f k. k Si At tim t = α j /1 + ɛ), Algorithm QG will construct an instanc of th Max-1-LFLP, i.., for ach clint l D and facility k F 1, dfin d kl = max{b l c lki, 0}. Not that B l r l,j /1 + ɛ) for l < j and B l = t for l j. Thn w considr a fasibl solution of th Max-1-LFLP. W opn facilitis in S i with opning cost at most k S i f k. For ach l D i, w assign l to th facility in S i, to which l is connctd in th optimal solution of th 2-LFLP. For othr clints not in D i, w assign thm to any opn facility. Th total profit of this solution is at last max{b l m l, 0} f k. l D i k S i By Thorm 1, Algorithm QG can find a solution for th Max-1-LFLP with total profit at last 1 1 ) l D i max{b l m l, 0} k S i f k. 10

11 Howvr, th total amount of offrs rcivd by i from th clints is at last this quantity that can not b gratr than th f i. Thrfor, 1 1 ) l D i max{b l m l, 0} k S i f k f i. Rarranging th trms in th abov inquality, w gt th dsird conclusion. Thorm 2. Th 2-LFLP can b approximatd by a factor of ɛ) 2 in polynomial tim for any givn constant ɛ > 0. Proof. By Lmma 2 and Lmma 4, by comparing systms 3) and 4), and by using th pair a, b) = 1.118, 1.77), w gt α j 1 + ɛ) f i + f k m i, 1 k Si j D i j D i which, togthr with Lmma 3, implis that Algorithm QG is a ɛ) 2 -approximation algorithm for th 2-LFLP sinc A local improvmnt procdur for th 2-LFLP Th main rsult of this sction is that givn a R f, R c )-approximation algorithm for th 2-LFLP, w can find an approximation algorithm with prformanc guarant R f + 1 lnδ), 1 + R c 1 δ ) for any δ 1 in polynomial tim. W will s its application in th analysis of algorithms for th 3-LFLP and th 4-LFLP, among othrs. A similar rsult has bn provd for UFLP by Guha and Khullr [19] and w hav sn many applications of it [10, 19, 26, 27, 5]. Th ky hr is again th quasi-grdy approach. In ordr to prov th abov rsult, w dsign a local improvmnt procdur for th 2-LFLP. Roughly spaking, onc w hav a fasibl solution for th 2-LFLP, w may add som facilitis to th currnt solution such that th total cost is rducd. Th local improvmnt procdur procds as follows, which is similar to Algorithm QG. W ar givn a solution for th 2-LFLP. Assum th connction cost of ach clint j D is o j. W want to add som facilitis opn on facility in F 2 and simultanously many facilitis in F 1 ) into th currnt solution. At ach stp, w considr an unopn facility i F 2. W construct an instanc of th Max-1-LFLP: w ar givn th st of clints D, th st of facilitis F 1, th facility cost for opning k F 1 is f k w can lt f k = 0 if k has bn opn in th currnt solution), and th rvnu gnratd by assigning j D to k F 1 is d jk = max{0, o j c jki }. Again, w solv th Max-1-LFLP by using Algorithm MAX. Assum that clint j is assignd to σ i j) by Algorithm MAX. If th total profit of th solution for th Max-1-LFLP is gratr than f i, thn th total cost of th currnt solution for th 2-LFLP can b rducd by opning facility i, and for ach clint j, 11

12 if d jσi j) > 0 thn opn σ i j) and r-connct j to th path σ i j), i). Such a stp is calld an add opration. Rpat this procdur until th solution can not b improvd by any add opration Considr any fasibl solution, say OP T, for th 2-LFLP. Assum its total connction cost and facility cost ar C opt and F opt. Again, w.l.o.g., w can assum that th solution OP T forms a forst. Considr on of th trs rootd at i F 2. Lt S i and D i b dfind as thos in th last sction. Now w ar rady to prov th following Lmma, whos countrpart for th 1-LFLP is wll-known in this ara. Lmma 5. If no mor add opration can improv th currnt solution, thn C C opt + 1 F opt. Proof. Lt OP T F i ) b th subst of facilitis in F i that ar opn in th fasibl solution OP T, for i = 1, 2. In th local improvmnt procdur, assum that w ar considring a candidat facility i OP T F 2 ). Lt th connction cost of clint j b, o j in th currnt solution, and θ j in OP T. As in th proof of Lmma 4, thr xists a fasibl solution for th Max-1-LFLP with total profit at last max{o j θ j, 0} f k. j D i k S i Thrfor, Algorithm MAX can find a solution with total profit at last 1 1 ) max{o j θ j, 0} f k 1 1 ) o j θ j ) f k. j D i k S i j D i k S i Howvr, sinc no add opration can improv th currnt solution, w must hav f i 1 1 ) o j θ j ) f k. j D i k S i Not that for i l, D i D l =, i OP T F2 )D i = D and i OP T F2 )F i 1 = OP T F 1). Thn w add th abov inquality ovr i OP T F 2 ), which complts th proof. Th abov lmma lads to th following conclusion immdiatly: Lmma 6. If thr is an a, b)-approximation algorithm for th 2-LFLP, thn ) w can gt an approximation algorithm with prformanc guarant a + b 1 1 1), 1 + for any 1. Proof. Assum that thr is an a, b)-approximation algorithm for 2-LFLP, thn by scaling th facility cost of th original instanc by a factor of 1, w can find a solution such that F + C a F OP T + bc OP T. Thn w apply th local improvmnt procdur on this solution, until th cost on th scald instanc) can not b rducd anymor. Thn w must hav, by Lmma 5, C C OP T + 1 F OP T. 12

13 Combining ths two inqualitis, w must hav F + C a + ) 1) F OP T b 1 1 )COP T. Thorm 3. For any givn ɛ > 0, if thr is an a, b)-approximation algorithm for th 2-LFLP, thn w can gt an approximation algorithm with prformanc guarant a + ) b 1 ln ) + ɛ, for any 1. Proof. Assum that w hav an a, b)-approximation algorithm for 2-LFLP. Lt δ 1. W prov that for any intgr p 1, thr xists an approximation algorithm for 2-LFLP with prformanc guarant a + b 1 1pδ 1), 1 + δ ). p W prov th claim by induction on p. Th cas p = 1 is trivial by Lmma 6. Assum that th claim is corrct for p 1. Thn w hav an a + p 1)δ 1) 1, 1 + )- b 1 δ p 1 approximation algorithm for 2-LFLP. W apply Lmma 6 again, thn w gt an approximation algorithm with prformanc guarant p 1)δ 1) δ 1) a , ) b 1 1 δ p 1 δ p, which is xactly what w nd. Thus w hav provd th claim for any intgr p. Now for any 1, lt δ = 1 p for som larg intgr p. W hav ) thus provd that any a, b)-approximation algorithm implis an a + 1 p 1 p 1), 1 + b 1 -approximation algorithm. Not that p 1 p 1) ln whn p. Thus, For any ɛ > 0, thr xists a constant p such that This complts th proof of th Thorm. p 1 p 1) ln + ɛ. 5 Variants of th 2-LFLP In this sction, w prsnt improvd rsults on approximating svral variants of th 2-LFLP. 13

14 5.1 Th non-mtric 2-LFLP W show that Algorithm QG has a prformanc guarant of Oln D )) for th non-mtric 2- LFLP. Th analysis is th sam as that of th mtric 2-LFLP, xcpt that in Lmma 4, on of th inqualitis of 4) dos not ncssarily hold. In fact, w can prov that Lmma 7. Lt f = 1 fi + k S i f k ), thn th systm of inqualitis 5) holds: 1 j < k : α j α j+1 1 j k : k l=j maxα j/1 + ɛ) m l, 0) f 1 j k : α j, m j, f 0. 5) This lads to th following thorm. Thorm 4. Th non-mtric 2-LFLP can b approximatd by a factor of Oln D )) in polynomial tim. Proof. Th inquality systm 5 implis that, for ach j : 1 j k: 1 α j f i + k f k + k j k Si It follows that whr H k = k 1 i k α j H k f i + f k + 1 k Si j=1 l=1 k l=1 m l lnk). This complts th proof of th thorm. To th bst of our knowldg, no approximation algorithm for th non-mtric 2-LFLP is known in th litratur. Hochbaum [20] has shown that th non-mtric 1-LFLP can b approximatd by a factor of ln D ), and by a rsult of Fig [16], it is th bst possibl unlss P=NP. Thrfor, our ratio for th non-mtric 2-LFLP is th bst possibl up to a constant. m l., 5.2 Th two-lvl concntrator location problm In th two-lvl concntrator location problm 2-LCLP), w hav th sam input as that of th 2-LFLP. Howvr, ach clint must b srvd by a first lvl opn facility only, and ach of th first lvl opn facilitis must b srvd by a scond lvl opn facility. To b prcis, w ar askd to choos subst = V t F t to opn, for t = 1, 2 such that min c jk + min c ki + k V 1 i V 2 k S 1 j D 2 f it t=1 i t V t is minimizd. Hr, w assum that th connction cost satisfy th triangl inquality. 14

15 As that for th 2-LFLP, w can assum that th opn paths of an optimal solution of th 2-LCLP form a forst. W considr any tr of th forst with its root i F 2. Again, w dnot th lavs of th tr by S i that is a subst of F 1. And D i is th st of clints that ar assignd to th tr rootd at i in th optimal solution. Thrfor, th total cost of th optimal solution) associatd with this tr is f i + f k + c ki ) + min c jk. k S i k S i j D i If w could prov that, for ach i F 2, α j R f f i + k + c ki ) + R c j D i k Sif j D i min k S i c jk, thn Algorithm QG must b a R f, R c )-approximation algorithm for th 2-LCLP. W can apply Algorithm QG to th 2-LCLP as wll. Th only chang w should mak is th construction of th instancs of th Max-1-LFLP. In particular, whn considring a facility i F 2, th instanc of th Max-1-LFLP is constructd as follows: th st of clints is D, th st of facilitis is F 1, th facility cost for opning facility k F 1 is f k +c ki, and th rvnu gnratd by assigning clint j to facility k is max{0, B j c jk }, whr B j = c jk if j is currntly connctd to a facility k F 1, othrwis B j = α j. Thn following th sam analysis as that for 2-LFLP, w can show that th 2-LCLP can b approximatd by a factor of ɛ) 2 in polynomial tim for any constant ɛ > 0. Lvin [23] claimd a constant factor approximation algorithm for th k-lclp whn k is a constant. Th approximation ratio of his algorithm is xponntial on k. Our rsult significantly improvs th ratio for k = Th 3- and th 4-LFLP In [5], improvd algorithms for th k-lflp hav bn proposd by rducing it to th 1-LFLP. In fact, in ordr to gt improvd ratios for th k-lflp, on nds to combin two rduction mthods: th paramtrizd path rduction and th rcursiv rduction. For dtails of th rductions, w invit th author to rfr to [5]. Sinc w hav a strong approximation for th 2-LFLP, w can furthr rfin thir rduction. W omit th proofs of Lmma 8 and Lmma 9, sinc complt proofs would ssntially rpat all th argumnts of [5]. For th 3-LFLP, w us xactly th sam rductions as thos in [5]. Howvr, whn w apply th rcursiv rduction to th 3-LFLP, w nd to solv an instanc of th 2-LFLP. In [5], th instanc of th 2-LFLP is furthr rducd to two instancs of th 1-LFLP. Instad of doing this, w now can solv th instanc of th 2-LFLP dirctly by using Algorithm QG. This lads to th following lmma. Lmma 8. Assum that th 1- and th 2-LFLP can b approximatd by factors of a, b) and α, β), rspctivly, thn th 3-LFLP can b approximatd by factors of max{a, a+α 3b+β 2 }, 2 ). Thrfor, w can obtain a bttr approximation ratio for th 3-LFLP. Th prviously bst known ratio is

16 Thorm 5. Th 3-LFLP can b approximatd by a factor of Proof. By Thorm 2 and Thorm 3, and by ltting = , w know that th 2-LFLP can b approximatd by a factor of α, β) such that α = / 1) ln )), β ) = / By Lmma 2, and by ltting δ = , w know that th 1-LFLP can b approximatd by a factor of a, b) = ln ), / ). It follows from Lmma 8 that th 3-LFLP can b approximatd by a factor of For th 4-LFLP, w modify th rductions of [5] in th following way. In th paramtrizd path rduction, for any instanc of th 4-LFLP, w rduc it to an instanc of th 2-LFLP. In th rcursiv path rduction, w rduc any instanc of th 4-LFLP to two instancs of th 2-LFLP in [5], it will b rducd to on instanc of th 1-LFLP, and on instanc of th 3-LFLP). W can prov th following lmma. Lmma 9. Assum that th 2-LFLP can b approximatd by a factor of α, β), thn th 4-LFLP can b approximatd by factors of α, 2β). Thorm 6. Th 4-LFLP can b approximatd by a factor of Proof. By Thorm 2 and Thorm 3, w know that th 4-LFLP can b approximatd by a factor of α, β) such that α ) = / 1) ln ), β ) = / for som 1. By ltting = 1.92, w hav α and β Thrfor, by Lmma 9, w gt a 2.81-approximation algorithm for th 4-LFLP. W rmark that no approximation ratio bttr than 3 was known for th k-lflp for any k Th 2-LFLP with soft capacity This problm is th sam as th 2-LFLP xcpt that th facility cost for opning facility i is a function of th numbr of clints it srvs. In particular, if i srvs k clints, thn th facility cost of i is f i k u i whr f i and u i ar givn. for ach facility i, thr is an uppr bound u i on th numbr of clints it can srv. It has bn shown in [5] Thorm 2, pag 154) that th 2-LFLP can b approximatd by a factor of 1.104, 3.56). By Thorm 3 and by ltting = 1.281, w obtain a 1.5, 3)-approximation algorithm for th 2-LFLP. W follow th approach of [27] and show that any α, β)-approximation algorithm for th 2-LFLP implis a 2α, 2β)-approximation algorithm for th 2-LFLP with soft capacity. Thrfor, w can gt a 3-approximation algorithm for th 2-LFLP with soft capacity. 16

17 6 Concluding Rmarks Srval qustions rmain opn. First of all, although our bound dos not dpnd on th LP rlaxation for th 2-LFLP [33], it would b of intrst to know th intgrality gap of th LP rlaxation. A rlatd qustion is on th lowr bound of th 2-LFLP, i.., whthr w can improv it from that is th known lowr bound for th 1-LFLP. Although in many applications of th multi-lvl problm, th numbr of lvls is small, it is crtainly of thortical intrst to study th approximability of th k-lflp for gnral k. Can our tchniqu b xtndd to th k-lflp for k 2? This rquirs a good approximation for th maximization vrsion of th k 1)-LFLP, which has bn studid in [6] and [35]. But thir rsults ar not in th form that can b applid within our framwork. Finally, our rsults, if combind with that of Aardal, Chuadk, and Shmoys [2] strongly suggst that it is possibl to dvlop a polynomial tim algorithm for th k-lflp whos prformanc guarant dpnds on k, i.., th prformanc guarant is strictly lss than 3 for any k, and convrgs to 3 whn k gos to infinity. Acknowldgmnt. Th author would lik to thank Yinyu Y for his support and his fdbacks on prliminary drafts of this papr. Th author also thanks Alxandr Agv, Asaf Lvin, Pt Vinott, Dachuan Xu and SODA 2004 program committ for thir hlpful commnts. Rfrncs [1] K. Aardal, M. Labb, J. Lung and M. Quyrann, On th two-lvl uncapacitatd facility location problm, INFORMS Journal on Computing, 8, , [2] K. Aardal, F.A. Chudak and D.B. Shmoys, A 3-approximation algorithm for th k-lvl uncapacitatd facility location problm, Information Procssing Lttrs, 72, , [3] A. Agv, Improvd approximation algorithms for multilvl facility location problms, Opr. Rs. Lttrs 30, , [4] A. Agv and M. Sviridnko, An approximation algorithm for uncapacitatd facility location problm, Discrt Applid Mathmatics, 93, , [5] A. Agv, Y. Y and J. Zhang, Improvd Combinatorial Apporixmation Algorithms for th k-lvl Facility Location Problm, Procdings of Th 30th Intrnational Colloquium on Automata, Languags and Programming ICALP), LNCS 2719, , [6] A. Bumb, An approximation algorithm for th maximization vrsion of th two lvl uncapacitatd facility location problm, Oprations Rsarch Lttrs, 294), , [7] A.F. Bumb and W. Krn, A simpl dual ascnt algorithm for th multilvl facility location problm, 4th Intrnational Workshop on Approximation Algorithms for Combinatorial Optimization APPROX 2001), LNCS 2129, 55-62,

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