We present a simple dual ascent method for the multilevel facility location problem which finds a solution within $6$ times the optimum for the uncapacitated case and within $12$ times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique. \u

Bumb, A.F.

Kern, W.

English

We present a simple dual ascent method for the multilevel facility location problem which finds a solution within $6$ times the optimum for the uncapacitated case and within $12$ times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique

Kern, Walter

NARCIS 

A simple dual ascent algorithm for the multilevel facility location problem

University of Twente Research Information

Faculty of Mathematical SciencesUniversity of TwenteUniversity for Technical and Social SciencesP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlMemorandum No. 1574A simple dual ascent algorithmfor the multilevel facility location problemA.F. Bumb and W. KernMarch 2001ISSN 0169-2690A simple dual ascent algorithm for the multilevel facility locationproblemA.F. Bumb and W.KernFaculty of Mathematical SciencesUniversity of TwenteMarch 9, 2001AbstractWe present a simple dual ascent method for the multilevel facility location problem which findsa solution within 6 times the optimum for the uncapacitated case and within 12 times the optimumfor the capacitated one. The algorithm is deterministic and based on the primal-dual technique.Keywords: approximation algorithms, linear programming, discrete locationAMS Subject Classifications: 68W25, 90C05, 90B801 IntroductionAn important problem in facility location is to select a set of facilities, such as warehouses or plants, inorder to minimize the total cost of opening facilities and of satisfying the demands for some commodity(see Cornuejols, Nemhauser & Wolsey [ CNW90] ). There are two models which are manly studied in theliterature: the uncapacitated facility location model (UFL), in which each facility can serve an unlimitednumber of clients and the capacitated facility location model in which each facility can serve at most acertain number of clients. We shall briefly survey the results known on approximation algorithms forthese two facility location problems.Throughout this paper, a ρ− approximation algorithm is a polynomial time algorithm that alwaysfinds a feasible solution with objective function value within ρ times the optimum. The value ρ is calledthe performance guarantee of the algorithm.It is known that the metric uncapacitated facility location problem (the service costs are symmetricand satisfy the triangle inequality) is NP-hard (see[ CNW90]). Independently, Guha &Khuller [ GK98]and Sviridenko [ S97] have shown that it is Max SNP-hard. They have also proved that the existence ofa ρ−approximation algorithm with ρ < 1.46 would imply that P = NP.The first constant-factor approximation algorithm for the metric uncapacitated facility location prob-lem was developed by Shmoys, Tardos & Aardal [ STA97], and was based on the filtering techniqueintroduced by Lin & Vitter[ LV92]. The performance guarantee of this algorithm is 3.16. Their resultwas subsequently improved by Guha & Khuller[ GK98], who obtained a 2.41-approximation algorithmapplying a greedy procedure to the solution obtained by [ STA97] and by Chudak & Shmoys [ C98], whoprovided an 1.73-approximation algorithm based on a randomized rounding technique, in which the vari-ables are rounded in a dependent way. For the capacitated version, Chudak & Shmoys [ CS99] obtaineda 3-approximation algorithm in the case that one can open at one location an infinite number of copiesof one facility. The drawback of these algorithms is that they need to solve large linear programs andtherefore they are not practical for many applications.Recently, Jain&Vazirani proposed the first combinatorial approximation algorithm, based on theprimal-dual technique, which achieves the performance guarantee of 3 for the uncapacitated version and14 for the capacitated one. Charikar & Guha [ CG99] observed that scaling the costs, running the primal-dual algorithm for the new problem and improving the solution using the greedy procedure from[ GK98]can reduce the performance guarantee to 1.85.The multilevel extension of the UFL problem has also been extensively studied in the literature (see[ A96]). In this problem there are k types of facilities to be built: one type of depots and (k − 1) typesof transit stations. Each unit of demand must be shipped from a depot through transit stations of typek − 1, ..., 1 to the demand points. Being an extension of UFL, this problem is Max SNP-hard as well.In [ STA97] Shmoys, Tardos & Aardal extended their algorithm for one level to multiple levels offacilities, obtaining a performance guarantee of 3.16, constant which was later improved to 3 by Aardal,Chudak & Shmoys [ ACS99]. As in the one level case, their algorithm is based on a randomized roundingof a LP solution to an integer one. The first combinatorial algorithm for the multilevel facility locationproblem was developed by Meyerson, Munagala&Plotkin [ MMP], and finds a solution within O (log |D|)the optimum, where D is the set of demand points.Using an idea from [ JV99], we present a simple greedy (dual ascent) method for the multilevel facilitylocation problem that finds a solution within 6 times the optimum. The algorithm can be easily extendedto the capacitated case, when each facility can serve only a certain number of demand points. Theapproximation guarantee for this case is 12.2 The metric multilevel uncapacitated facility location problemConsider a complete (k+ 1)−partite graph G = (V,E) with V = V0 ∪ ...∪Vk and E =k⋃l=1Vl−1 × Vl. Theset D = V0 is the set of demand nodes and F = V1 ∪ ... ∪ Vk is the set of possible facility locations (atlevel 1, ..., k). We are given edge costs c ∈ RE+ and opening costs f ∈ RF+ ( i.e., opening a facility at i ∈ Fincurs a cost fi ≥ 0). We assume that c is induced by a metric on V . Without loss of generality we canassume that there are no edges of cost 0.REMARK Our results also hold in a slightly more general setting, where we require only fore ∈ V0 × V1 that c(e) ≤ c(p) for any path p joining the endpoints of e.Denote by P the set of paths of length k−1 joining some node in V1 to some node in Vk. If j ∈ D andp = (v1, ..., vk) ∈ P, we let jp denote the path (j, v1, ..., vk). As usual c(p) and c(jp) denote the length ofp resp. jp ( with respect to c).The corresponding UFL problem can now be stated as follows: Determine for each j ∈ D a pathpj ∈ P ( along ”open facilities”) so as to minimize∑j∈Dc(jpj) + f(⋃j∈Dpj).REMARK In this setting we assume that each j ∈ D has a demand of one unit to be shipped alongpj . Our results easily extend to arbitrary positive demands.To derive an integer programming formulation of the UFL problem, we introduce the 0− 1 variablesyi (i ∈ F ) to indicate whether i ∈ F is open and the 0 − 1 variables xjp (j ∈ D, p ∈ P ) to indicatewhether j is served along p.We letc(x) :=∑p∈P∑j∈Dcjpxjpandf(y) :=∑i∈Ffiyi.2The UFL problem is now equivalent tominimize c(x) + f(y)subject to∑p∈Pxjp = 1, for each j ∈ D (1)(Pint)∑p3ixjp ≤ yi, for each i ∈ F, j ∈ D (2)xpj ∈ {0, 1} , for each p ∈ P, j ∈ Dyi ∈ {0, 1} , for each i ∈ FConstraints (1) ensure that each j gets connected via some path and constraints (2) ensure that thepaths only use open facilities.The LP−relaxation of (Pint) is given byminimize c(x) + f(y)(P ) subject to (1), (2)xjp ≥ 0yi ≥ 0.Note that xjp ≤ 1 is implied by (1) and yi ≤ 1 holds automatically for any optimal solution (x, y) of(P ).The standard way of proving a 0− 1 solution (x, y) of (Pint) to be a ρ−approximation is to show thatc(x) + f(y) ≤ ρCLP (2.1)where CLP is the optimum value of (P ).3 The primal-dual algorithmThe basic idea of the primal-dual approach is to exhibit a primal 0− 1 solution (x, y) satisfying (2.1) byconsidering the dual of (P ). Introducing dual variables vj and tij corresponding to constraints (1) and(2) in (P ), the dual becomesmaximize∑j∈Dvjvj −∑i∈ptij ≤ c(jp), for each p ∈ P , j ∈ D (3)∑j∈Dtij ≤ fi, for each i ∈ F (4)tij ≥ 0, for each i ∈ F , j ∈ DIntuitively, the dual variable vj indicates how much j ∈ D is willing to pay for getting connected.The value of tij indicates how much j ∈ D is willing to contribute to the opening cost fi ( if he would beconnected along a path through i).We aim at constructing a primal feasible 0− 1 solution (x, y) and a feasible dual solution (v, t) suchthatc(x) + f(y) ≤ 6∑j∈Dvj ,3implying (2.1) for ρ = 6.We first describe how to construct the dual solution (v, t) . To this end,we introduce the followingnotation w.r.t. an arbitrary feasible solution (v, t) of (D) :A facility i ∈ F is fully paid when ∑j∈Dtij = fi. (3.1)A demand point j ∈ D reaches il ∈ Vl if for some path p = (i1, ..., il) from V1to il all facilities i1, ...il−1are fully paid andvj = cjp +∑i∈ptij . (3.2)If, in addition, also il is fully paid, we say that j leaves il or, in case l = k, that j gets connected (along p to ik ∈ Vk).Our algorithm for constructing the dual solution is a dual ascent method, generalizing the approachin [ JV99]. We start with v ≡ t ≡ 0 and increase all vj uniformly ( ”with unit speed” ). When somej ∈ D reaches a not fully paid node i ∈ F, we start increasing tij with unit speed, until fi is fully paidand j leaves i.We stop increasing vj when j gets connected . The algorithm maintains the invariant thatat time T the dual variables vj that are still being raised are all equal to T.More precisely, we proceed asdescribed below.UNTIL all j ∈ D are connected DO• Increase vj for all j ∈ D not yet connected• Increase tij for all i ∈ F, j ∈ D satisfying (i)− (iii) :(i) j has reached i(ii) j is not yet connected(iii) i is not yet fully paid .Let (v, t) denote the final dual solution. Before constructing a corresponding primal solution (x, y),let us state a few simple facts about (v, t).For each fully paid facility i ∈ Vl , l ≥ 2,denote by Ti the time when facility i became fully paid.Thepredecessor of i will be the facility in the level l−1 via which i was for the first time reached by a demandpoint, i.e.,Pred (i) =i′ ∈ Vl−1| i′ is fully paid and Ti′ + ci′i = mini′′∈Vl−1i′′fully paid(Ti′′ + ci′′i) .( Ties are broken arbitrarily.)The predecessor of a fully paid facility i ∈ V1 will be its closest demand point. We can define thetime TPred(i) = 0.For all fully paid facilities i in the k − th level denote by ji pi = (i1, ...ik) the path through thefollowing points:• ik = i• il =Pred(il+1) , for each 1 ≤ l ≤ k − 1•ji =Pred(i1) .We will call the neighborhood of i the set of demand nodes contributing to pi i.e.,Ni = {j ∈ D | ti′j > 0 for some i′ ∈ pi} .Since each j ∈ D gets connected we may fix for each j ∈ D a connecting path p̃j ∈ P of fully paidfacilities ( ties are broken arbitrarily).4LEMMA 3.1.(i) c(jp̃j) ≤ vj for all j ∈ D(ii) For all j ∈ D and i ∈ Vk fully paid such that i ∈ p̃j ,either vj = Ti and tij > 0 or vj > Ti andtij = 0(iii) For all fully paid facilities i ∈ Vk and corresponding paths pi = (i1,..., ik), the following relationholdsTi1 ≤ ... ≤ Tik(iv) Let i ∈ Vk be a fully paid facility and pi = (i1,..., ik) its asoociated path. For all j ∈ D and il ∈ piwith tilj > 0, there exists a path p from V1to il such thatc (jp) +k−1∑s=lcisis+1 ≤ Ti.In particular, c(jipi) ≤ Ti(v) If i, i′ are two fully paid facilities in Vk with intersecting neighborhoods then for each j′ ∈ D, suchthat i′ ∈ p̃j′ , cjij′ ≤ 4 max {Ti, vj′}(vi)∑i′∈piti′j ≤ vj for all j ∈ DProofThe first claim is straightforward from (3.2) and the definition of p̃j.The second claim is based on the observation that at time T all the v−values that can be increasedare equal with T and that the final v−values reflect the times when the demand points get connected.There are two possibilities that a fully paid facility i ∈ Vk is on the connecting path of a demand pointj. One is that j reached i before Ti and got connected when i became fully paid .In this case tij > 0 andvj = Ti. The other possibility is that j reached i after i was fully paid, which means that tij = 0 andvj > Ti.The definition of a predecessor implies that for each fully paid i ∈ FcPred(i)i + TPred(i) ≤ Ti. (3.3)The third claim follows immediately .For the forth claim, by adding the inequalities (3.3) for il+1, ...ik−1 one obtainsk−1∑s=lcisis+1 + Til ≤ Tik .Since tilj > 0, there is a path p along which j reached il before Til . Clearly, c(jp) ≤ Til , which implies(iv).For proving (v), let j ∈ Ni ∩Ni′ . Since j ∈ Ni, there is an il ∈ pi such that tilj > 0 . Then by (iv),there exists a path q from V1 to il such that c (jq) ≤ Ti.Suppose pi′ = (i′1, ..., i′k) . Similarly, there is an i′r ∈ pi′ and a path q′ from V1 to i′r such thatc (jq′) +k−1∑s=rci′si′s+1 ≤ Ti′ .Using the triangle inequality and (ii), we obtaincjij′ ≤ c(jipi) + c(jq) + c(jq′) +k−1∑s=rci′si′s+1 + c(j′p̃j′)≤ 2Ti + Ti′ + vj′≤ 2Ti + 2vj′≤ 4 max {Ti, vj′} .5Finally, for proving the statement in the last claim is enough to show that no demand point j couldincrease simultaneously two values tilj , tisj , for il 6= is and il, is ∈ pi. This follows from the definition ofpi,which implies that whenever a demand point reaches a facility on pi, the predecessor of that facilityshould have been already paid, and subsequently all the facilities of pi situated on inferior levels.We now describe how to construct a corresponding primal solution (x, y) . Suppose there are r fullypaid facilities in the last level. Order them according to nondecreasing T−values, sayT1 ≤ ...Tr.Construct greedily a set C ⊆ Vk of centers which have parewise disjoint neighborhoods and assigneach j ∈ D to some center i0 ∈ C as follows:INITIALIZE C = ∅FOR i = 1, ...r DOIF Ni ∩Ni0 6=  for some i0 ≤ i, assign to pi0 all demand nodesj ∈ D with i ∈ p̃jELSE C = C ∪ {i} and assign to pi all the demand nodes j ∈ Dwith the property that i ∈ p̃jThe paths pi (i ∈ C) are called central paths.Remark Note that each demand point j is assigned to one center. Furthermore, by construction ofC, j ”contributes” to at most one central path ( not necessarily the one to which it is assigned).The primal solution (x, y) is obtained by connecting all demand nodes along their correspondingcentral paths:xjp :={1 if p = pi and j was assigned to pi0 otherwiseandyi :={1 if i is on a central path0 otherwise .The shipping cost c (x) is easily bounded as follows.If j ∈ D is assigned to pi0 then Ti0 ≤ Ti,where {i} = p̃j ∩ Vk. Due to Lemma 3.1 (ii) and (v), we getTi0 ≤ vj andcjpi0 ≤ cji0 j + cji0pi0 ≤ 4vj + Ti0 ≤ 5vj .The cost of opening facilities along a central path pi0 can be also bounded with the help of Lemma3.1(vi) ∑i∈pi0fi =∑i∈pi0∑j∈Nitij ≤∑j∈Nivj .Since the centers have pairwise disjoint neighborhoods, we further conclude thatf(y) =∑i0∈C∑i∈pi0fi ≤∑j∈Dvj .6We have proved .THEOREM 3.1 The above primal solution (x, y) satisfiesc(x) + f(y) ≤ 6∑j∈Dνj .3.1 A capacitated versionThe following capacitated version has been considered in the literature (cf. [ GMM00]): Each i ∈ F hasan associated node capacity ui ∈ N which is an upper bound on the number of paths using i. On theother hand, we are allowed to open as many copies of i ( at cost fi each) as needed.To formulate this, we replace the 0−1 variables yi in (Pint) by nonnegative integer variables yi ∈ Z+,indicating the number of open copies of i ∈ F. Furthermore, we add capacity constraints∑j∈D∑p3ixjp ≤ uiyi, for each i ∈ F. (4.1)Again, we let CLP denote the optimum value of the corresponding LP−relaxation.The idea to approach the capacitated case (also implicit in [ JV99] for the 1-level case) is to move thecapacity constraints to the objective using Lagrangian multipliers λi ≥ 0, for each i ∈ F. This results inan uncapacitated problemC(λ) := minimize c(x) + f(y) +∑i∈Fλi∑j∈D∑p3ixjp − uiyi= minimize c̃(x) + f̃(y)with f̃i = fi − λiui, for each i ∈ F and c̃(e) = c(e) + λi if i is the endpoint of e ∈ E. Note that eachλ ≥ 0 gives C(λ) ≤ CLP .As in section 3, we compute a primal 0− 1 solution (x, y) of C(λ) withc̃(x) + f̃(y) ≤ 6C(λ).Note that this does not necessarily satisfy the capacity constraints (4.1). However, a clever choice ofthe Lagrangian multipliers λi = 12fiui(i ∈ F ) yieldsc̃(x) + f̃(y) = c(x) +12∑i∈Ffiui∑p3i∑j∈Dxjp +12∑i∈Ffiyi≥ c(x) + 12∑i∈Ffiyi,where yi :=⌈1ui∑p3i∑j∈Dxjp⌉opens each facility i ∈ F sufficiently many times. Hence (x, y) is indeeda feasible solution of the capacitated problem satisfyingc(x) +12f(y) ≤ 6C(λ) ≤ 6CLP ,7hencec(x) + f(y) ≤ 12CLP .Theorem 4.1 Our greedy dual ascent method yields a 12−approximation of the multilevel capacitatedfacility location problem.References[ A96] K. Aardal On the two-level uncapacitated facility location problem. INFORMS Journal onComputing, 8, pages 289-301, 1996[ ACS99] K.I.Aardal, F.Chudak and D.B.Shmoys. A 3-approximation algorithm for the k-level unca-pacitated facility location problem. Information Processing Letters, 72, pages 161-167, 1999[ C98] F. Chudak. Improved approximation algorithms for the uncapacitated facility location. In:R.E. Bixby, E.A. Boyd, R.Z. Rios-Mercado (eds.), Integer Programming and CombinatorialOptimization, 6th International IPCO Conference. Lecture Notes in Computer Science 1412,pages 180-194, Springer-Verlag, berlin Heidelberg, 1998[ CNW90] G. Cornuejols, G. L. Nemhauser and L. A. Wolsey. The uncapacitated facility location prob-lem. In P. Mirchandani and R. Francis, editors, Discrete Location Theory, pages 119-171. JohnWiley and Sons, New York, 1990[ GK98] S. Guha and S. Khuller. Greedy strikes back: improved facility algorithms. Proceedings of the9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 649-657, 1998[ KPR98] M. R. Korupolu, C. G. Plaxton and R. Rajaraman. Analysis of a local search heuristic forfacility location problems. Technical Report 98-30, DIMACS, June 1998[ JV99] K.Jain and V.V.Vazirani. Primal-dual approximation algorithms for metric facility locationand k−median problems. Proceedings of the Twenty-Ninth Annual IEEE Symposium on Foun-dations of Computer Science, 1999[ LV92] J. H. Lin and J. S. Vitter. ε−approximation with minimum packing constraint violation.Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 771-782,1992[ MMP] A. Meyerson, K. Munagala, S. Plotkin. Cost distance: Two metric network design. Proceedingsof the 41-th IEEE Symposium on Foundation of Computer Science, 2000[ STA97] D. B. Shmoys, E. Tardos and K. I. Aardal. Approximation algorithms for facility locationproblems. Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages265-274, 1997[ CG99] M. Charikar and S. Guha. Improved combinatorial algorithms for facility location and k-median problems. Proceedings of the Twenty-Ninth Annual IEEE Symposium on Foundationsof Computer Science, 1999[ GMM00] S. Guha, A. Meyerson, K. Munagala. Hierarchical Placement and Network Design Problems.to appear in Proceedings of the Thirtyth Annual IEEE Symposium on Foundations of Com-puter Science,2000[ CS99] F. Chudak and D. Shmoys. Improved approximation algorithms for the capacitated facility lo-cation problem. Proceedings of the tenth ACM-SIAM Symposium on Discrete Algorithms,1999[ S97] M. Sviridenko, Personal communication, 19978

We present a simple dual ascent method for the multilevel facility location problem which finds a solution within 6 times the optimum for the uncapacitated case and within f 2 times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual techiniqu

A simple dual ascent algorithm for themultilevel facility location problemAF Bumb WKernFaculty of Mathematical Sciences University of Twente PO Box   AEEnschede The Netherlandsemail afbumb	mathutwentenlkern	mathutwentenlAbstractWe present a simple dual ascent method for the multilevel facility location problemwhich nds a solution within  times the optimum for the uncapacitated case andwithin  times the optimum for the capacitated one The algorithm is deterministicand based on the primaldual techinique IntroductionThis paper concernes with the multilevel facility location problem Being anextension of the uncapacitated facility location problem which is known tobe Max SNPhard see this problem is Max SNPhard as well The 	rstapproximation algorithms for the multilevel facility location problem were developed by Shmoys Tardos  Aardal  and Aardal Chudak  Shmoys  andwere based on rounding of an LP solution to an integer one The performanceguarantee of these algorithms were  respectively  The 	rst combinatorial algorithm for the multilevel facility location problem was developed byMeyerson MunagalaPlotkin  and 	nds a solution within O log jDj theoptimum where D is the set of demand points Using an idea from  wepresent a simple greedy dual ascent method for the multilevel facility location problem that 	nds a solution within  times the optimum The algorithmextends to a capacitated variant of the problem when each facility can serveonly a certain number of demand points with an increase of the performanceguarantee to Preprint submitted to Elsevier Science  April  The metric multilevel uncapacitated facility location problemConsider a complete k  partite graph G  VE with V  V   Vkand E kSlVl  Vl The set D  V is the set of demand nodes andF  V    Vk is the set of possible facility locations at level   k Weare given edge costs c  RE and opening costs f  RF  ie opening a facilityat i  F incurs a cost fi   We assume that c is induced by a metric onV  Without loss of generality we can assume that there are no edges of cost Denote by P the set of paths of length k   joining some node in V tosome node in Vk If j  D and p  v  vk  P we let jp denote the pathj v  vk As usual cp and cjp denote the length of p resp jp  withrespect to cThe corresponding UFL problem can now be stated as follows Determinefor each j  D a path pj  P  along open facilities so as to minimizePjDcjpjfSjDpj In this setting we assume that each j  D has a demandof one unit to be shipped along pj  Our results easily extend to arbitrarypositive demandsTo derive an integer programming formulation of the UFL problem we introduce the   variables yi i  F  to indicate whether i  F is open and the  variables xjp j  D p  P  to indicate whether j is served along p Welet cx PpPPjDcjpxjp and fy PiFfiyiThe UFL problem is now equivalent tominimize cx  fysubject toPpP xjp   for each j  D Ppi xjp  yi for each i  F  j  D xjp yi  f g for each p  P  j  D i  FDenote with P  the LPrelaxation of the above integer program obtainedby allowing x y to be positive reals The dual ascent algorithmIntroducing dual variables vj and tij corresponding to constraints  and in P  the dual of P  can be written asmaximizePjDvjsubject to vj Piptij  cjp for each p  P  j  D PjDtij  fi for each i  F tij   for each i  F  j  DWe 	rst describe how to construct the dual solution v t  To this endweintroduce the following notation wrt an arbitrary feasible solution v t ofD  A facility i  F is fully paid whenPjDtij  fi A demand point j  Dreaches il  Vl if for some path p  i  il from Vto il all facilities i ilare fully paid and vj  cjp Piptij If in addition also il is fully paid we saythat j leaves il or in case l  k that j gets connected  along p to ik  VkOur algorithm for constructing the dual solution is a dual ascent methodgeneralizing the approach in We start with v  t   and increase all vj uniformly  with unit speed Until all demand points get connected we proceed as follows When somej  D reaches a not fully paid node i  F we start increasing tij with unitspeed until fi is fully paid and j leaves iWe stop increasing vj when j getsconnected The algorithm maintains the invariant that at time T the dualvariables vj that are still being raised are all equal to TLet v t denote the 	nal dual solution For each fully paid facility i  Vl l  denote by Ti the time when facility i became fully paidThe predecessorof i will be the facility in the level l   via which i was for the 	rst timereached by a demand point iePredi i Vlj iis fully paid and Ti  cii  miniVli fully paidTi  ciiTies are broken arbitrarily The predecessor of a fully paid facility i  Vwill be its closest demand point We can de	ne the time TPredi  For all fully paid facilities i in the k  th level denote by ji pi  i ik thepath through the following points ik  i il Predil  for each   l  k  ji Predi We will call the neighborhood of i the set of demand nodes contributing to piie Ni  fj  D j tij   for some i pi g Since each j  D gets connectedwe may 	x for each j  D a connecting path fpj  P of fully paid facilities ties are broken arbitrarily We now describe how to construct a correspondingprimal solution x ySuppose there are r fully paid facilities in the last level Order them accordingto nondecreasing Tvalues say T  Tr Construct greedily a set C 	 Vk ofcenters which have parewise disjoint neighborhoods and assign each j  Dto some center i  C in the following way For each fully paid facility i in thelast level check if there is a center i  i such that Ni  Ni   If such ani exist assign to pi all demand nodes j  D with i  fpj  Otherwise add i toC and assign to pi all the demand nodes j  D with i  fpj The primal solution x y is obtained as followsxjp  if p  pi and j was assigned to i  Vk otherwiseandyi  if i is on a central path otherwiseTheorem  The above primal solution x y satisescx  fy  XjDj implying that the performance guarantee of algorithm is NOTE The greedy dual ascent algorithm combined with a technique based onLagrangian multipliers yields a  approximation algorithm for the capacitatedvariant of the multilevel facility location problem in which each facility canserve only a certain number of demand pointsReferences KIAardal	 FChudak and DBShmoys A approximation algorithm for theklevel uncapacitated facility location problem Information Processing Letters		 pages 	  SGuha and S Khuller Greedy strikes back improved facility algorithmsProceedings of the th Annual ACMSIAM Symposium on Discrete Algorithmspages 	  KJain and VVVazirani Primaldual approximation algorithms for metricfacility location and kmedian problems Proceedings of the TwentyNinthAnnual IEEE Symposium on Foundations of Computer Science  A Meyerson	 K Munagala	 S Plotkin Cost distance Two metric networkdesign Proceedings of the th IEEE Symposium on Foundation of ComputerScience  D B Shmoys	 E Tardos and K I Aardal Approximation algorithms for facilitylocation problems Proceedings of the th Annual ACM Symposium on Theoryof Computing pages  

We present a simple dual ascent method for the multilevel facility location problem which finds a solution within 6 times the optimum for the uncapacitated case and within 12 times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique

Bumb, Adriana

A Simple Dual Ascent Algorithmfor the Multilevel Facility Location ProblemAdriana Bumb and Walter KernFaculty of Mathematical Sciences, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlandsa.f.bumb,kern@math.utwente.nlAbstract. We present a simple dual ascent method for the multilevelfacility location problem which finds a solution within 6 times the op-timum for the uncapacitated case and within 12 times the optimum forthe capacitated one. The algorithm is deterministic and based on theprimal-dual technique.1 IntroductionAn important problem in facility location is to select a set of facilities, such aswarehouses or plants, in order to minimize the total cost of opening facilitiesand of satisfying the demands for some commodity (see Cornuejols, Nemhauser& Wolsey [CNW90]).In this paper we consider the multilevel facility location problem in whichthere are k types of facilities to be built: one type of depots and (k − 1) typesof transit stations. For every type of facility the opening cost is given. Eachunit of demand must be shipped from a depot through transit stations of typek−1, . . . , 1 to the demand points. We assume that the shipping costs are positive,symmetric and satisfy the triangle inequality. The goal of the problem is to selectfacilities of each type to be opened and to connect each demand point to a pathalong open facilities such that the total cost of opening facilities and of shippingall the required demand from depots to demand points is minimized.Being an extension of the uncapacitated facility location problem, which isknown to be Max SNP-hard (see [GK98] and [S97]), this problem is Max SNP-hard as well. The first approximation algorithms for the multilevel facility loca-tion problem were developed by Shmoys, Tardos & Aardal [STA97] and Aardal,Chudak & Shmoys [ACS99] and were based on rounding of an LP solution toan integer one. The performance guarantees of these algorithms were 3.16, re-spectively 3. The first combinatorial algorithm for the multilevel facility locationproblem was developed by Meyerson, Munagala & Plotkin [MMP00], and findsa solution within O (log |D|) the optimum, where D is the set of demand points.Using an idea from [JV99], we present a simple greedy (dual ascent) methodfor the multilevel facility location problem that finds a solution within 6 timesthe optimum. The algorithm extends to a capacitated variant of the problem,when each facility can serve only a certain number of demand points, with anincrease of the performance guarantee to 12.M. Goemans et al. (Eds.): APPROX-RANDOM 2001, LNCS 2129, pp. 55–63, 2001.c© Springer-Verlag Berlin Heidelberg 200156 Adriana Bumb and Walter Kern2 The Metric Multilevel Uncapacitated FacilityLocation ProblemConsider a complete (k+1)−partite graph G = (V,E) with V = V0∪. . .∪Vk andE =k⋃l=1Vl−1×Vl. The set D = V0 is the set of demand nodes and F = V1∪. . .∪Vkis the set of possible facility locations (at level 1, . . . , k). We are given edge costsc ∈ RE+ and opening costs f ∈ RF+ ( i.e., opening a facility at i ∈ F incurs acost fi ≥ 0). We assume that c is induced by a metric on V . Without loss ofgenerality we can assume that there are no edges of cost 0.Remark 1. Our results also hold in a slightly more general setting, where werequire only for e ∈ V0×V1 that c(e) ≤ c(p) for any path p joining the endpointsof e.Denote by P the set of paths of length k − 1 joining some node in V1 tosome node in Vk. If j ∈ D and p = (v1, . . . , vk) ∈ P , we let jp denote the path(j, v1, . . . , vk). As usual c(p) and c(jp) denote the length of p resp. jp (withrespect to c).The corresponding facility location problem can now be stated as follows:Determine for each j ∈ D a path pj ∈ P (along ”open facilities”) so as tominimize ∑j∈Dc(jpj) + f(⋃j∈Dpj).Remark 2. In this setting we assume that each j ∈ D has a demand of one unitto be shipped along pj . Our results easily extend to arbitrary positive demands.To derive an integer programming formulation of the multilevel facility lo-cation problem, we introduce the 0 − 1 variables yi (i ∈ F ) to indicate whetheri ∈ F is open and the 0 − 1 variables xjp (j ∈ D, p ∈ P ) to indicate whether jis served along p.We letc(x) :=∑p∈P∑j∈Dcjpxjpandf(y) :=∑i∈Ffiyi .A Simple Dual Ascent Algorithm 57The multilevel facility location problem is now equivalent tominimize c(x) + f(y)subject to∑p∈Pxjp = 1, for each j ∈ D (1)(Pint)∑pixjp ≤ yi, for each i ∈ F, j ∈ D (2)xpj ∈ {0, 1} , for each p ∈ P, j ∈ Dyi ∈ {0, 1} , for each i ∈ FConstraints (1) ensure that each j gets connected via some path and con-straints (2) ensure that the paths only use open facilities.The LP−relaxation of (Pint) is given byminimize c(x) + f(y)(P ) subject to (1), (2)xjp ≥ 0yi ≥ 0 .Note that xjp ≤ 1 is implied by (1) and yi ≤ 1 holds automatically for anyoptimal solution (x, y) of (P ).The standard way of proving a 0 − 1 solution (x, y) of (Pint) to be a ρ−approximation is to show thatc(x) + f(y) ≤ ρCLP (2.1)where CLP is the optimum value of (P ).3 The Primal-Dual AlgorithmThe basic idea of the primal-dual approach is to exhibit a primal 0 − 1 solution(x, y) satisfying (2.1) by considering the dual of (P ). Introducing dual variablesvj and tij corresponding to constraints (1) and (2) in (P ), the dual becomesmaximize∑j∈Dvjvj −∑i∈ptij ≤ c(jp), for each p ∈ P , j ∈ D (3)∑j∈Dtij ≤ fi, for each i ∈ F (4)tij ≥ 0, for each i ∈ F , j ∈ DIntuitively, the dual variable vj indicates how much j ∈ D is willing to payfor getting connected. The value of tij indicates how much j ∈ D is willing to58 Adriana Bumb and Walter Kerncontribute to the opening cost fi (if he would be connected along a path throughi).We aim at constructing a primal feasible 0 − 1 solution (x, y) and a feasibledual solution (v, t) such thatc(x) + f(y) ≤ 6∑j∈Dvj ,implying (2.1) for ρ = 6.We first describe how to construct the dual solution (v, t). To this end, weintroduce the following notation w.r.t. an arbitrary feasible solution (v, t) of (D):A facility i ∈ F is fully paid when∑j∈Dtij = fi. (3.1)A demand point j ∈ D reaches il ∈ Vl if for some path p = (i1, . . . , il) fromV1to il all facilities i1, . . . il−1 are fully paid andvj = cjp +∑i∈ptij . (3.2)If, in addition, also il is fully paid, we say that j leaves il or, in case l = k,that j gets connected (along p to ik ∈ Vk).Our algorithm for constructing the dual solution is a dual ascent method,generalizing the approach in [JV99]. We start with v ≡ t ≡ 0 and increase allvj uniformly ( ”with unit speed” ). When some j ∈ D reaches a not fully paidnode i ∈ F , we start increasing tij with unit speed, until fi is fully paid and jleaves i. We stop increasing vj when j gets connected. The algorithm maintainsthe invariant that at time T the dual variables vj that are still being raised areall equal to T . More precisely, we proceed as described below.UNTIL all j ∈ D are connected DO• Increase vj for all j ∈ D not yet connected• Increase tij for all i ∈ F , j ∈ D satisfying (i) − (iii),(i) j has reached i(ii) j is not yet connected(iii) i is not yet fully paid.Let (v, t) denote the final dual solution. Before constructing a correspondingprimal solution (x, y), let us state a few simple facts about (v, t).For each fully paid facility i ∈ Vl, l ≥ 2, denote by Ti the time when facilityi became fully paid. The predecessor of i will be the facility in the level l− 1 viawhich i was for the first time reached by a demand point, i.e.,Pred (i) =i′ ∈ Vl−1| i′ is fully paid and Ti′ + ci′i = mini′′∈Vl−1i′′fully paid(Ti′′ + ci′′i) .A Simple Dual Ascent Algorithm 59(Ties are broken arbitrarily.)The predecessor of a fully paid facility i ∈ V1 will be its closest demandpoint. We can define the time TPred(i) = 0.For all fully paid facilities i in the k − th level denote by ji pi = (i1, . . . , ik)the path through the following points:• ik = i• il =Pred(il+1), for each 1 ≤ l ≤ k − 1•ji =Pred(i1).We will call the neighborhood of i the set of demand nodes contributing to pii.e.,Ni = {j ∈ D | ti′j > 0 for some i′ ∈ pi} .Since each j ∈ D gets connected we may fix for each j ∈ D a connectingpath p̃j ∈ P of fully paid facilities (ties are broken arbitrarily).Lemma 1. (i) c(jp̃j) ≤ vj for all j ∈ D(ii) For all j ∈ D and i ∈ Vk fully paid such that i ∈ p̃j, either vj = Ti andtij > 0 or vj > Ti and tij = 0(iii) For all fully paid facilities i ∈ Vk and corresponding paths pi = (i1, . . . ,ik), the following relation holdsTi1 ≤ . . . ≤ Tik(iv) Let i ∈ Vk be a fully paid facility and pi = (i1, . . . , ik) its associated path.For all j ∈ D and il ∈ pi with tilj > 0, there exists a path p from V1to il suchthatc (jp) +k−1∑s=lcisis+1 ≤ Ti .In particular, c(jipi) ≤ Ti(v) If i, i′ are two fully paid facilities in Vk with intersecting neighborhoodsthen for each j′ ∈ D, such that i′ ∈ p̃j′ , cjij′ ≤ 4 max {Ti, vj′}(vi)∑i′∈piti′j ≤ vj for all j ∈ DProof. The first claim is straightforward from (3.2) and the definition of p̃j .The second claim is based on the observation that at time T all the v−valuesthat can be increased are equal with T and that the final v−values reflect thetimes when the demand points get connected. There are two possibilities that afully paid facility i ∈ Vk is on the connecting path of a demand point j. One isthat j reached i before Ti and got connected when i became fully paid. In thiscase tij > 0 and vj = Ti. The other possibility is that j reached i after i wasfully paid, which means that tij = 0 and vj > Ti.The definition of a predecessor implies that for each fully paid i ∈ FcPred(i)i + TPred(i) ≤ Ti . (3.3)60 Adriana Bumb and Walter KernThe third claim follows immediately.For the forth claim, by adding the inequalities (3.3) for il+1, . . . , ik−1 oneobtainsk−1∑s=lcisis+1 + Til ≤ Tik .Since tilj > 0, there is a path p along which j reached il before Til . Clearly,c(jp) ≤ Til , which implies (iv).For proving (v), let j ∈ Ni ∩ Ni′ . Since j ∈ Ni, there is an il ∈ pi such thattilj > 0. Then by (iv), there exists a path q from V1 to il such that c (jq) ≤ Ti.Suppose pi′ = (i′1, . . . , i′k). Similarly, there is an i′r ∈ pi′ and a path q′ fromV1 to i′r such that c (jq′) +k−1∑s=rci′si′s+1 ≤ Ti′ .Using the triangle inequality and (ii), we obtaincjij′ ≤ c(jipi) + c(jq) + c(jq′) +k−1∑s=rci′si′s+1 + c(j′p̃j′)≤ 2Ti + Ti′ + vj′≤ 2Ti + 2vj′≤ 4 max {Ti, vj′} .Finally, for proving the statement in the last claim is enough to show that nodemand point j could increase simultaneously two values tilj , tisj , for il = is andil, is ∈ pi. This follows from the definition of pi, which implies that whenever ademand point reaches a facility on pi, the predecessor of that facility should havebeen already paid, and subsequently all the facilities of pi situated on inferiorlevels. We now describe how to construct a corresponding primal solution (x, y).Suppose there are r fully paid facilities in the last level. Order them accordingto nondecreasing T−values, sayT1 ≤ . . . ≤ Tr .Construct greedily a set C ⊆ Vk of centers which have parewise disjointneighborhoods and assign each j ∈ D to some center i0 ∈ C as follows:INITIALIZE C = ∅FOR i = 1, . . . , r DOIF Ni ∩ Ni0 =  for some i0 ≤ i, assign to pi0 all demand nodesj ∈ D with i ∈ p̃jELSE C = C ∪ {i} and assign to pi all the demand nodes j ∈ Dwith the property that i ∈ p̃jThe paths pi (i ∈ C) are called central paths.A Simple Dual Ascent Algorithm 61Remark 3. Note that each demand point j is assigned to one center. Further-more, by construction of C, j ”contributes” to at most one central path (notnecessarily the one to which it is assigned).The primal solution (x, y) is obtained by connecting all demand nodes alongtheir corresponding central paths:xjp :={1 if p = pi and j was assigned to pi0 otherwiseandyi :={1 if i is on a central path0 otherwise .The shipping cost c (x) is easily bounded as follows.If j ∈ D is assigned to pi0 then Ti0 ≤ Ti,where {i} = p̃j ∩Vk. Due to Lemma1 (ii) and (v), we get Ti0 ≤ vj andcjpi0 ≤ cji0 j + cji0pi0 ≤ 4vj + Ti0 ≤ 5vj .The cost of opening facilities along a central path pi0 can be also boundedwith the help of Lemma 1(vi)∑i∈pi0fi =∑i∈pi0∑j∈Nitij ≤∑j∈Nivj .Since the centers have pairwise disjoint neighborhoods, we further concludethatf(y) =∑i0∈C∑i∈pi0fi ≤∑j∈Dvj .We have provedTheorem 1. The above primal solution (x, y) satisfiesc(x) + f(y) ≤ 6∑j∈Dνj .4 A Capacitated VersionThe following capacitated version has been considered in the literature: Eachi ∈ F has an associated node capacity ui ∈ N which is an upper bound on thenumber of paths using i. On the other hand, we are allowed to open as manycopies of i (at cost fi each) as needed.62 Adriana Bumb and Walter KernTo formulate this, we replace the 0 − 1 variables yi in (Pint) by nonnega-tive integer variables yi ∈ Z+, indicating the number of open copies of i ∈ F .Furthermore, we add capacity constraints∑j∈D∑pixjp ≤ uiyi, for each i ∈ F . (4.1)Again, we let CLP denote the optimum value of the corresponding LP -relaxation.The idea to approach the capacitated case (also implicit in [JV99] for the 1-level case) is to move the capacity constraints to the objective using Lagrangianmultipliers λi ≥ 0, for each i ∈ F . This results in an uncapacitated problemC(λ) := minimize c(x) + f(y) +∑i∈Fλi∑j∈D∑pixjp − uiyi= minimize c̃(x) + f̃(y)with f̃i = fi − λiui, for each i ∈ F and c̃(e) = c(e) + λi if i is the endpoint ofe ∈ E. Note that each λ ≥ 0 gives C(λ) ≤ CLP .As in section 3. we compute a primal 0 − 1 solution (x, y) of C(λ) withc̃(x) + f̃(y) ≤ 6C(λ) .Note that this does not necessarily satisfy the capacity constraints (4.1).However, a clever choice of the Lagrangian multipliers λi = 12fiui(i ∈ F ) yieldsc̃(x) + f̃(y) = c(x) +12∑i∈Ffiui∑pi∑j∈Dxjp +12∑i∈Ffiyi≥ c(x) + 12∑i∈Ffiyi ,where yi :=⌈1ui∑pi∑j∈Dxjp⌉opens each facility i ∈ F sufficiently many times.Hence (x, y) is indeed a feasible solution of the capacitated problem satisfyingc(x) +12f(y) ≤ 6C(λ) ≤ 6CLP ,hencec(x) + f(y) ≤ 12CLP .Theorem 2. Our greedy dual ascent method yields a 12−approximation of themultilevel capacitated facility location problem.A Simple Dual Ascent Algorithm 63ReferencesACS99. 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A simple dual ascent algorithm for the multilevel facility location problem

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