Sngle Snk Buy at Bulk Problem and the Access Network

Transcription

1 A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buy-at-bulk network desgn problem, where we have to desgn a network by buyng ppes of dfferent costs and capactes per unt length to route demands at a set of sources to a sngle snk. The dstances n the underlyng network form a metrc. Ths result mproves the prevous bound of O(log R ), where R s the set of sources. We also present a better constant approxmaton to the related Access Network Desgn problem. Our algorthms are randomzed and combnatoral. As a subroutne n our algorthm, we use an nterestng varant of faclty locaton wth lower bounds on the amount of demand an open faclty needs to serve. We call ths varant load balanced faclty locaton, and present a constant factor approxmaton for t, whle relaxng the lower bounds by a constant factor. Ths paper combnes the work n two conference papers [12, 13], whch appeared n the 41st IEEE Symposum on the Foundatons of Computer Scence, 2000 and the 33rd ACM Symposum on Theory of Computng, 2001 respectvely. Department of Computer and Informaton Scences, Unversty of Pennsylvana. Emal: sudpto@cs.upenn.edu. Research supported by an NSF CAREER award and a Sloan Foundaton Fellowshp. Department of Computer Scence, Unversty of Calforna, Los Angeles CA Emal: awm@cs.ucla.edu Department of Computer Scence, Duke Unversty, Durham NC Emal: kamesh@cs.duke.edu. Research supported by NSF va a CAREER award and grant CNS

2 1 Introducton Network desgn problems requre layng cables on an underlyng metrc n order to connect a set of demand ponts. The network must support each demand pont operatng at a known peak (or average) rate, and we would lke the cheapest possble network supportng these demands. In a metrc scenaro (whch s standard), f the cost of cables s lnear n the amount of bandwdth they provde, ths problem s polynomal-tme solvable usng multcommodty flow technques. However, n several real applcatons the costs of cables obey economes of scale; the cost-per-unt-bandwdth s less for a hgh-capacty cable. Smlar problems arse outsde the data networks communty, e.g., n locaton theory where we consder transportng products for sale where the cost s a concave functon of the amount of demand transported. The concavty also may arse mplctly, e.g., n clusterng data where we are wllng to tolerate a less dense cluster f t contans a larger number of ponts. The problem of buy-at-bulk network desgn, wth a sngle snk node to whch all the demand has to be routed, was frst ntroduced by Salman et al n [21]. They gave a O(mn{log n, log D}) approxmaton, where the number of nodes s n and D s the maxmum demand. They also showed that the problem s HP-hard. Awerbuch and Azar [3] whch gave an O(log 2 n) approxmaton for the multple snk case, where dfferent demand ponts may communcate wth dfferent snks. Ths result may be mproved to O(log n) usng subsequent results of [4, 5, 9] on approxmaton of metrcs usng trees. Andrews and Zhang [2] consdered the Access Network Desgn problem, where all demands need to connect to a sngle conceptual entty, the core of the network whch represents the nternet backbone, a set of fle servers, or the factores where a product s produced. Even though there may exst multple snks, the snks are symmetrc n that demand ponts do not care to whch snk they are connected. The access network desgn problem s a specal case of sngle snk buy at bulk problem wth the added twst that cables have to be utlzed up-to a mnmum capacty. All of the above problems can be termed as an unform varant, n the sense that any cable type can be used between any par of vertces. For the more general non-unform problem where cable types may have lmted avalablty (but the network stll has a sngle snk) Meyerson et al provded an O(log n) approxmaton n [19]. In ths paper we focus on the unform sngle snk case. Our man contrbutons are: Our algorthm provdes the frst constant factor approxmaton for the sngle-snk buy-at-bulk problem. Ths subsumes the Access network desgn problem [2] as well. The approxmaton rato we obtan for the buy-at-bulk problem s 292. We provde a Structure Theorem that allows us to dentfy the key regons n the concave functon. Ths allows one to wrte an LP wth O(1) ntegralty gap. All subsequent analyss of ths problem showng a O(1) approxmaton (see below and Secton 7) depend on ths theorem as well. We defne and provde the frst constant factor bcrtera approxmaton algorthm for a natural varant of faclty locaton, the Load Balanced Faclty Locaton problem where there s a lower bound specfed on the demand an open faclty needs to serve. Ths problem arses n several clusterng scenaros as well [14]. We provde randomzed combnatoral algorthms, but they can be derandomzed usng standard technques. The paper [12] contaned several other results on mult-level network desgn problems whch were based on smlar technques. We omt ther dscusson n the nterest of keepng the presentaton focused and smple. 1

3 Soluton Technque: There are several mportant novel deas n our soluton methodology. Frst, the optmal soluton could use many cables wth slghtly dfferng costs n successon, obtanng margnal beneft from each. Such a soluton s hard to characterze. We show that to a constant factor approxmaton, we can replace such smlar cable types wth the same cable type. Ths leads to a soluton wth smpler layered characterzaton (Structure Theorem). In partcular, we show that a near-optmal soluton s composed of alternatng layers of Stener and shortest path forests, each layer usng only a sngle type of cable. The next layer gets used only when suffcent demand has been accumulated at the roots of the prevous layer to make the next cable type cost-effectve. Such a soluton effectvely herarchcally aggregates demand n order to explot the economes of scale n cable costs. The challenge now s to approxmate the layered soluton. We show that the shortest path nstance corresponds to a varant of the well-studed faclty locaton problem, where a certan mnmum demand (correspondng to makng the next cable type feasble) must be collected at the facltes. We present a constant factor approxmaton for ths varant, whch we term Load-balanced faclty locaton. Ths s an ndependent contrbuton of ths work. We present a smple constant factor approxmaton for the Stener forest varant as well. We fnally show that an teratve bottomup aggregaton of demand usng these Stener and shortest path forests yelds a constant factor approxmaton. Independent Results: Independent of ths work, Karger and Mnkoff [16] defned the load balanced faclty locaton problem as a subroutne for solvng the sngle commodty Rent-or-Buy problem. They obtaned the same algorthm as the one we present n Secton 6. We note that the sngle commodty Rent-or-Buy problem s a specal case of the sngle snk buy-at-bulk problem consdered n ths paper. Garg et al [10] obtaned a O(K) approxmaton (where K s the number of dfferent cable types) to the sngle snk buy-at-bulk problem by roundng the natural LP formulaton. Our paper mproves the approxmaton rato va a fully combnatoral algorthm. Snce the natural LP relaxatons for the Stener forest and shortest path problems have constant ntegralty gap, t s natural to expect that an LP based on the structure theorem should have O(1) ntegralty gap as well, and we provde ths LP for completeness. However, the authors of [10] observe that the structure theorem n our paper can be used on ther stronger, natural LP, as well. Ths connecton has also been made explct by Talwar [23] subsequently. Organzaton of the Paper: In Secton 2, we state the sngle snk buy-at-bulk problem formally, and dscuss the structural propertes of the optmal soluton. In Secton 3.2, we dscuss a scalng dea to remove smlar ppe types, and show how t mproves the structure of the optmum soluton. We then present the Herarchy algorthm n Secton 4 and show a constant approxmaton rato. We show n Secton 5 how to mprove the approxmaton rato for Access Network Desgn. We present the algorthm for load balanced faclty locaton n Secton 6, and conclude by surveyng results that appeared subsequent to the publcaton of a prelmnary verson of ths paper [13]. 2 Defntons and Prelmnares The Sngle Snk Buy at Bulk problem as defned n [21]: Gven a graph G(V, E) wth a dstance (length) functon c e on the edges, the goal s to construct a network routng a set S 1 V of demand nodes to a sngle snk s. We are gven K types of connectons (ppes) where ppe type has a fxed cost σ per unt length and a capacty u. Each demand node v S 1 needs to transport some amount of demand d v to the snk. The objectve s to optmze the cost of buyng ppes along the 2

4 edges to route all demands to the snk. We are allowed to buy multple copes of a ppe along the same lnk. The above can be termed as a capactated verson, we wll use an alternate ncremental cost formulaton of the above problem, whch easly models arbtrary pecewse lnear concave costs. Defnton 2.1 In the sngle snk buy at bulk problem we are gven a set of ppes, where ppe type has fxed and ncremental costs σ, δ respectvely. If we transport d unts of demand along a path of length L usng ppe type, we wll pay a total of L(σ + δ d). The goal s to construct a network routng a set S 1 V of demand nodes to a sngle snk s, whle mnmzng the cost of the network. Let f (Y ) = σ + δ Y to be the per-unt-dstance cost of routng demand Y along a ppe of type. Defne h(y ) = mn f (Y ). The functon h() s pecewse lnear and concave. Note that a ppe type whch does not affect h() (that s, do not defne the envelope) wll not affect the soluton at all. Therefore, f we only focus on the ppe types whch are useful, and number the ppes n decreasng order of σ we observe that δ 1 > δ 2 > > δ K. In ths formulaton the capacty of ppe k s u = σ /δ. It s not hard to see that a soluton under ths ncremental cost formulaton costs at least as much as the same soluton under the capactated model, and at most twce as much as the soluton under the capactated model. Furthermore, Lemma 2.1 [2] In the ncremental cost model the optmum soluton naturally defnes a tree. The above s straghtforward snce the costs are sub-addtve under the above assumptons of {σ }, {δ }; after the lnks n the optmum are bought and the fxed cost pad, the entre demand from each node can be routed along the path wth the lowest ncremental cost (tes broken arbtrarly) from each node to the snk. Ths would defne a tree, and we can elmnate the edges we do not use. Note that as a consequence, every node would use an unque ppe type for ts outgong flow. Also due to the subaddtvty property of the costs, along every flow path the ppe types wll ncrease n number. The authors of [2] also ntroduced the followng problem: Defnton 2.2 The Access Network Desgn problem s defned as follows: It s the same as the sngle snk buy at bulk problem wth the followng added restrctons (c = 1/2 s used n [2]) 1. For 2 k K, f d < c σ k δ k, then dδ k 1 + σ k 1 < dδ k + σ k. 2. The smallest demand looks lke the smallest ppe capacty, or more precsely, δ 1 > cσ κ<k σ κ = O(σ k ). As mentoned earler, our solutons for the above problems wll use solutons to the followng varant of the Faclty Locaton problem: Defnton 2.3 The Load Balanced Faclty Locaton problem s defned as: We are gven a network G(V, E) wth a dstance functon c( ) on the edges and a set of demand ponts, wth demands d j. The cost of openng a faclty at locaton s f. In addton, there s a lower bound of L on the demand a faclty opened at must satsfy. We are requred to open facltes and allocate the demands to the open facltes so that an open faclty at has at least L demand routed to t. The cost of our soluton s the sum of the dstances traveled by the demands and the cost of the open facltes. The goal s to mnmze ths cost. 3

5 3 Sngle Snk Buy at Bulk 3.1 Roadmap Intuton: Assume that we only consder the ppe types whch are not domnated by others. Thus the cost per unt length s a pecewse concave functon of the demand. Observe that as we ncrease demand along an edge, there are break-ponts at whch t becomes cheaper to use the next larger ppe type. Let g k be the demand for whch t becomes cheaper to use a ppe of type k + 1 compared to a ppe of type k. Suppose that we are n a scenaro 0 = g 0 < u 1 < g 1 < u 2 < g 2 < < u K < g K = (we wll show how to acheve a smlar scenaro later). Observe now that f the demand amount s n the range [g 1, u ], we can gnore the ncremental cost wth a factor 2 loss n cost, and the cost of the edge wll just be σ tmes the length of the edge, ndependent of the demand. If on the other hand, the demand s n the range [u, g ], we can gnore the fxed cost wth a factor 2 loss n cost, and the cost of the edge per unt length s δ tmes the demand. Ths mples that the optmum soluton can be converted wth a factor 2 loss n cost to a layered soluton. Layer has a Stener forest usng ppes of type followed by a forest of shortest path trees usng ppes of the same type. Each ppe n the Stener forest has at least g 1 demand and each ppe n the shortest path forest has at least u amount of demand. The shortest path forest should ensure that we collect at least a demand of g, such that we can use the Stener forest correspondng to a larger ppe type. Ths gves us a clusterng problem where each cluster s supposed to have a mnmum number of (weghted by demand) ponts ths s the reason for usng the load balanced faclty locaton problem. For completeness, we frst defne the faclty locaton problem [22]: Defnton 1 (Faclty Locaton) We are gven a set of demands D. Let d j be the demand at j D. We are gven a set of feasble locatons F, where φ s the cost of openng a faclty at locaton F. The ponts D and F are embedded n a metrc space where c j s the dstance between ponts and j. The goal s to open a subset of facltes X F and connect each demand j D to the closest open faclty q(j) X, so that the total cost of the open facltes, X φ plus the sum of the routng cost, j D c q(j)jd j s mnmzed. Let ρ f denote the best approxmaton rato for the faclty locaton problem. Ths s 1.52 due to [18]. The Load Balanced Faclty Locaton problem has an added constrant: Each F has a lower bound L on the demand t needs to serve f opened. The soluton X constructed must satsfy that for every X, at least L demand s routed to n the soluton. In Secton 6 we prove the followng: Theorem 3.1 We can compute a soluton for load balanced faclty locaton whose cost s 2ρ f tmes that of the optmal soluton, such that our soluton relaxes the lower bounds by a factor of 1/3, so that for X, at least L /3 demand s routed to. However, f we apply the above drectly and compare ourselves wth the optmal soluton that satsfes the above mentoned structural propertes. the analyss does not mmedately go through. So we use the dea that we wll only use a larger ppe type when t s sgnfcantly,.e., by a constant factor, cheaper. Ths wll allow us to set up a geometrc seres that accounts for the cost of the shortest path forests. But to bound the cost of the Stener forests we wll need a dfferent dea, namely we restrct ourselves to ppes where the σ decrease by a constant factor as well. But ths now mples that we should show that even after rulng out ppes accordng to the above two deas, there s a feasble soluton whch s not too expensve. Ths s the structure theorem we prove. 4

6 There s however one remanng ssue regardng how to correlate the costs of the dfferent layers we ntroduce a novel strategy where the entre demand of a layer s sent to random node n S 1. Note that t s mportant for our analyss that the demands be sent to S 1, because the structured soluton we derve from modfyng the optmum soluton s defned wth these demand nodes as the ground truth. In the remander of the secton, we frst descrbe the structured feasble soluton n Secton 3.2. We then present the algorthm n Secton Constructng a Layered Soluton We now formalze the ntuton descrbed above to obtan a layered soluton wth cost close to the optmal cost. Our algorthm wll progressvely construct partal solutons usng each ppe type n turn. In order to bound the total cost, we must guarantee that ppes are very dfferent from one another n terms of fxed and ncremental costs. Defnton 3.1 Defne a set of ppe types to be good f for some α (0, 1/2) we have: 1. For any k < K, we have σ k < ασ k For any k < K, we have αδ k > δ k+1. We need to prove that we can guarantee these condtons wthout ncreasng the cost of the optmum soluton by too much. Lemma 3.1 There exsts a set of good ppes and a soluton that uses only these types, such that the cost of ths soluton s at most 1/α tmes the cost of the orgnal optmum soluton. Proof: We frst elmnate ppes n order to guarantee that among the remanng ppes we have σ k < ασ k+1 whle ncreasng the fxed cost of the optmum soluton by at most 1/α. The ncremental cost of the optmum soluton can only decrease durng ths phase. We fnd the largest ppe k such that σ k ασ k+1. We elmnate ths ppe, replacng t n the optmum soluton wth ppe k + 1. We renumber the ppes and repeat. Notce that f at some pont some ppe type s replaced by ppes of type k, then we wll always keep ppes of type k n the fnal soluton (snce every ppe type hgher than k has at least α hgher fxed cost). When ths fnshes, we wll have the desred property. The orgnal optmum soluton wth ppe replacements has fxed cost at most 1/α larger snce any ppe whch was replaced was replaced by a ppe wth at most 1/α bgger fxed cost. The ncremental cost can only decrease, snce hgher fxed cost mples smaller ncremental cost. We now elmnate ppes n reverse order, where an elmnated ppe s replaced by a ppe wth larger ncremental cost, to guarantee that among the remanng ppes we have αδ k > δ k+1 whle ncreasng the ncremental cost of the optmum soluton by at most 1/α. The fxed cost of the optmum soluton can only decrease. Combnng these two phases gves the soluton clamed by the lemma. Defnton 3.2 Assumng that we have a good set of ppes, defne b k to be such that f k+1 (b k ) = 2αf k (b k ). In essence, b k s suffcent demand that t becomes consderably cheaper to use a ppe of type k + 1 rather than a ppe of type k. 5

7 Lemma 3.2 For all k, u k b k u k+1. Proof: From the defnton of b k, we can wrte: σ k+1 + δ k+1 b k = 2α(σ k + δ k b k ).. Solvng ths equaton for b k yelds: b k = σ k+1 2ασ k 2αδ k δ k+1 σ k+1 2αδ k δ k+1 σ k+1 δ k+1 = u k+1 The above shows b k u k+1, to see the other bound observe that when we have b k flow, t s cheaper to use a ppe of type k + 1 rather than a ppe of type k. It follows that σ k+1 + δ k+1 b k < σ k + δ k b k. Solvng ths for b k, we can see that b k > σ k+1 σ k δ k δ k+1 Snce α < 1/2, t follows that σ k+1 > 2σ k and we can conclude that b k > u k. Lemma 3.3 For all k and any demand D b k, f k+1 (D) 2αf k (D). Proof: Suppose D = b k + x for some x 0. Then, f k+1 (D) = σ k+1 + δ k+1 (b k + x) = 2α(σ k + δ k b k ) + δ k+1 x. Notng that δ k+1 αδ k, t mmedately follows that f k+1 (D) 2αf k (D). Lemma 3.4 For all k and any demand D u k, f k+1 (D) f k (D). Proof: Note that f k+1 (D), f k (D) are nondecreasng lnear functons n D and further, δ k+1 δ k. Therefore, to prove the lemma, t suffces to observe that f k+1 (u k ) f k (u k ). σ But f k (u k ) = σ k + δ k k δ k = 2σ k < σ k /α σ k+1 f k+1 (u k ). Note that we requre α (0, 1 2 ). We now show that there exsts a structured near-optmum soluton. Subsequently we wll search for solutons whch obey ths structure and are wthn constant factor of the best structured soluton. Theorem 3.2 (Structure Theorem) There exsts a tree soluton that uses ppes of type k on a lnk ff the demand x on the lnk satsfes x [b k 1, b k ). Further, the tree routes all demand whch entered a node usng ppe k out of that node usng a ppe of type k or k + 1. Ths soluton pays at 1 most tmes the optmum soluton. 2α 2 Proof: Frst, as noted n Secton 2, the optmal soluton defnes a tree due to the sub-addtve nature of the costs. Further, as noted there, the ncremental cost model forces ths soluton to use only one ppe type per edge. Both these propertes wll be preserved by the transformatons descrbed below. We frst modfy the soluton to only use the set of good ppes (accordng to Lemma 3.1). Therefore we need to show that transformatons n the the rest of the proof ncreases the cost of the soluton by at most a factor of 1 2α. Consder any edge where x unts of flow s routed by the optmum soluton. Let k 0 = argmn f (x). Ths s the ppe type used by the optmum soluton. Suppose a ppe has flow b k 1 x < u k. We know by (repeated) applcaton of Lemmas 3.3 and 3.2 that a ppe of type k would reduce the cost compared to any smaller ppe type. Lkewse, by 6

8 (repeated) applcaton of lemmas 3.4 and 3.2 we can conclude that a ppe of type k would reduce the cost compared to any larger ppe type. Thus k = k 0. Therefore, what remans to be shown s that f u k x < b k and we use a ppe of type k, then our cost does not ncrease sgnfcantly. Frst, due to the dscusson above, snce b k u k+1, we know that t s cheaper to use ppe type k+1 compared to any larger ppe type. Thus k 0 = k or k 0 = k+1 and we need to compare f k (x) and f k+1 (x) only. Now by Defnton 3.2, and the fact that f k (x) and f k+1 (x) are lnear non-decreasng wth δ k+1 δ k ; t s mmedate that 2αf k (x) f k+1 (x). Thus f we modfy the optmum soluton (already restrcted to good ppes) to use ppe type k n the range [b k 1, b k ), then the cost of the soluton goes up by at most a factor of 1/(2α); combned wth Lemma 3.1 the total cost s at most 1/(2α 2 ) tmes the optmal cost. To acheve the second part of the lemma, we observe that snce we are consderng a tree soluton, the flow does not decrease as we proceed toward the root. We smply ntroduce dummy nodes or ppes of length 0, f the largest ncomng ppe type s k and the outgong ppe type s larger than k + 1. An LP Formulaton: We can encode the structural observaton above nto an nteger program formulaton. We modfy the graph to nclude K self-loops of length 0 at every vertex, n order to accomodate ppe types requred by the Structure Theorem. Denote by x vek whether the demand at node v uses a ppe of type k on edge e. By y ek we denote whether there exsts a ppe of type k on edge e. The nteger program can then be formulated as follows. Here In(v) denotes the set of edges comng nto on node v, and Out(v) the set gong out of v. Recall that c e s the length of edge e, and d v s the demand at node v S 1. Mnmze c e e E k σ k y ek + e δ k d v x vek v S 1 k e In(w) x vek = e Out(w) (x vek + x vek+1 ) v S 1, w V \ {s}, k x vek y ek v S 1, e E, k e Out(v) x ve1 = 1 v S 1 x vek, y ek {0, 1} The LP s obtaned by relaxng the fnal ntegralty constrants. It can be shown usng Theorem 4.1 that f the LP s wrtten on the set of good ppe types, t has an O(1) ntegralty gap. Drect LP roundng technques exst as well [10, 23]. Snce the LP s not the man focus of the paper, we omt a proof of the ntegralty gap. 4 The Herarchy Algorthm We wll now present the Herarchy algorthm for sngle snk buy-at-bulk based on the structural observatons we made above. The scalng dea from the prevous secton measures that we can compare the cost of our soluton n each layer wth the respectve costs of the optmum soluton. The algorthm s presented below: 7

9 Algorthm Herarchy Let s denote the snk node and S 1 the set of orgnal demand nodes. Assume s S 1. Let the demand of v S 1 {s} be denoted by d v. The algorthm proceeds n phases. In phase we wll use ppes of type only. Let D (v) denote the demand of a node v at the start of the phase. Snce the algorthm s randomzed, ths s a random varable. Let S be the set of non-zero demand ponts (and s) we have at ths stage. () Stener Trees: Construct an approxmately optmal Stener tree on S. Root ths tree at s. Transport the demands from S upwards along the tree. If on any edge, the amount of demand s larger than u, we cut the tree at that edge. Ths gves us a forest on S where each edge has at most u demand through t. () Consoldate A: Consder any subtree t n the above forest whch does not contan s. Let the set of nonzero demand nodes n t be S A(t). Pck a node y at random from SA (t) n proporton to ts demand D (y). For all nodes n S A (t), we send ther demand (whch are currently located at the root of t) to z usng ppes of type. Let A be the set of nodes y chosen correspondng to dfferent t. Denote the demand of a node v mmedately after ths step to be D A(v). () Shortest Path Trees: Approxmately solve a load balanced faclty locaton nstance on S 1 wth the faclty lower bound b on all nodes (and no faclty costs). If there does not exst b total demand, then we nstead route drectly to the snk. We get a forest of shortest path trees. We route our current demands along these trees to ther roots. Note that we solve the load balanced faclty locaton problem on the S 1 nodes and not on the S A (t) nodes. (v) Consoldate B: Consder any faclty p opened n the above forest of shortest path trees. Some set of nodes from S 1 were assgned to p, denoted by S B (p), and ther (orgnal) total demand s at least b /3. We choose a node z at random from S B (p) wth probablty proportonal to d z. For all y S B(p) A, we send ther demand (whch s currently at node p) to node z usng ppes of type. Let S +1 be the set of nodes z that are chosen correspondng to the dfferent facltes p. Note that the only nodes currently havng non-zero demands are nodes n S +1. Our soluton wll route the demands through the forests of ncreasng ppe types. Ths soluton need not be a tree, but can easly be converted to one of no greater cost. Let ρ s and ρ f denote the best approxmaton ratos for the Stener tree and faclty locaton problems respectvely. Note that ρ s = 1.55 due to [20] and ρ f = 1.52 due to [18]. Note that for the Shortest Path tree part, we use Theorem 3.1 to obtan a 2ρ f approxmaton that routes at least b /3 demand to each open faclty. For the Stener Trees part, we use the ρ s approxmaton. 4.1 Analyss Let Γ be the structured optmal soluton constructed n Theorem 3.2. We defne C to be the total cost of Γ usng ppes of type. The total cost of the structured optmal soluton s therefore K=1 C = C. Let T I be the ncremental cost of the Stener Tree at layer and T F cost of the Stener Tree at layer s T = T I + T F Let P I be the ncremental cost of the shortest path tree at layer and P F The total cost of the shortest path tree at layer s P = P I + P F.. be ts fxed cost. The total be ts fxed cost. Let N be the total cost of the consoldaton steps for layer. The total cost of our soluton s therefore (T + P + N ). 8

10 Lemma 4.1 For all, v, we have E[D (v)] = d v and E[D A (v)] = d v, that s the expected demand at any node after any of the consoldaton steps s the orgnal demand of the node. Proof: We wll prove ths by nducton on the steps. Suppose that the statement s true at some step. We wll show that t s true at the next step. There are two cases to consder; ether we performed a Stener Tree step or a Shortest Path Tree step. Note D 1 (v) = d v. For the ease of notaton, defne D A 0 (v) = d v for all nodes v. These two equatons defne the base case. Suppose we have just performed a Stener Tree step. By the nducton hypothess we know E[D (v)] = d v. The node v s a part of some tree t wth total demand D t. We then choose a node for consoldaton and the probablty that we choose node v s D (v)/d t. If we choose v, demand D t wll be placed there; otherwse the demand s 0. Thus the expected amount of demand at v, condtoned on the prevous 1 steps s D (v). If we now remove the condtonng, E[D A(v)] = E[D (v)] = d v as desred. Suppose we have just performed a Shortest Path Tree step. Note that we used the S 1 nodes for the load balanced faclty locaton constructon n ths step. The probablty we consoldate to v s d v /D(p, ) where v s assgned to p n ths stage and the total demand assgned to p s D(p, ). Note that D(p, ) = u S B (p) d u The demand of v s D +1 (v) after ths step. Note that v collects the demands of the nodes n set A S B(p); further the demand of a node u n ths ntersecton s DA (u). u S B(p) DA (u). Now Condtoned on the prevous steps, the expected value of D +1 (v) s f we remove the condtonng, E[D +1 (v)] = d v D(p, ) E[ Ths proves the nductve case. u S B (p) D A (u)] = d v D(p, ) u S B (p) E[D A (u)] = d v D(p,) d v D(p, ) u S B (p) d u = d v Lemma 4.2 E[N ] T + P. Proof: The proof essentally follows by observng that the cost s a concave functon of demand, and the consoldaton process pcks a random node n proporton to the demand sent to the root. Ths s obvous for the shortest path trees let the nodes that belongng to a tree rooted at faclty p be v 1,..., v r wth dstances l 1,..., l r from node p. Then the expected consoldaton cost s r va /D)f (D)l a p a=1(d r f (d va )l a = P p a=1 The nequalty follows snce f s a concave functon. For the other step, consder the consoldaton process to recursvely choose from the root a subtree wth probablty proportonal to the total demand n the subtree. We just saw the proof for a 1 level tree that the consoldaton cost s less than the cost of sendng the flows to the root. Ths argument s now repeated n each of the subtrees. Lemma 4.3 T I T F and P F P I. 9

11 Proof: Snce we cut the tree at any edge wth more than u k demand along t, we guarantee that the fxed cost pad on any edge we actually use exceeds the ncremental cost. By the same argument, the Stener Tree stage guarantees at least u k demand or zero everywhere. For the shortest path tree, f an edge has zero demand flowng on t, we wll pay zero for that edge. Otherwse there s at least u k demand on the edge and we pay an ncremental cost whch exceeds the fxed cost for the shortest path trees. Lemma 4.4 E[P I] 2ρ j= f j=1 α j Cj. Proof: Suppose the demands at the sources were those from S 1. Then one possble soluton would be Γ tself untl ppes of type +1 were used. We know that Γ must gather the desred b flow before usng ppes of type + 1. Snce we wll always pay the ncremental cost δ, and the ncremental costs scale by α, we can guarantee a total cost of at most j= j=1 α j Cj for ths soluton. Our actual demand at each node has expected value equal to the orgnal demand, so the expected value of a feasble soluton for P I s bounded as above. The extra factor s due to the approxmaton of the load balanced faclty locaton problem as stated n Theorem 3.1. Lemma 4.5 Pr[D (v) 0] 3d v /b 1. Proof: Let v be the chosen node (denoted by z n Step (v) of the algorthm) correspondng to some p. We obtan the nodes S by solvng an nstance of the load balanced faclty locaton problem on S 1 wth lower bounds b 1. In ths soluton, each node n S except s has demand at least b 1 /3. In other words u S B (p) d u b 1 /3. Now note that v s chosen ndependent of {D 1 (u)} and ths s the reason why S 1 s used and not S. Note that at any node u n S 1, the currently accumulated before the shortest path tree step s D 1 A (u). We therefore have: E[D (v) D (v) > 0] = u S B (p) E[D A 1(u)] = Now combned wth the fact that E[D (v)] = d v, the lemma follows. u S B (p) d u b 1 /3 Lemma 4.6 Recall ( ρ s be the approxmaton rato the Stener tree approxmaton algorthm used, then E[T F ] ρ j=k s j= α j Cj + ) j= 1 j=1 3(2α) j Cj. Proof: To bound the cost of T F, we wll show that there exsts a Stener tree Γ whch connects all S wth a low cost. Note that ths Stener tree wll have an approxmaton rato at most ρ s relatve to the optmal Stener tree on the set S. The tree Γ wll be the tree correspondng to the structured soluton Γ, where (1) all the ppes of type j use ppes of type and (2) for the nodes v wth D (v) > 0 the edges n the path to the root where any ppe of type j < s used, that correspondng edge now uses a ppe of type. The cost accordng to part (1) can be bounded by j=k j= α j Cj. Ths s because for any ppe of type j, we have σ α j σ j. Further, the fxed cost of usng the ppe of type j s less than ts total cost and thus the above bound follows. To bound the contrbuton of (2), we focus on the subtrees of Γ whch has total demand at most b 1 (and therefore use ppes of type j < ). Note that due to the ntroducton of zero length edges, these subtrees may share ther (sub)roots but they wll be edge dsjont. For an edge e, 10

12 let the demand flowng through t be x e < b 1 and suppose that Γ uses a ppe of type j e < for ths edge. Let the set of nodes n Γ below e be denoted by Γ e. Let the length of e be l e. Now the edge e s used n Γ f any v n the subtree below t has D (v) > 0. Therefore the probablty e s used s bounded above by v Γ e 3d v /b 1 usng Lemma 4.5 and the unon bound. Thus we pay a cost v Γ e 3d v b 1 l e σ = 3x e b 1 σ l e 3x e b 1 f (b 1 ) 3x e b 1 (2α) je f je (b 1 ) 3(2α) je f je ( x e b 1 b 1 ) = 3(2α) je f je (x e ) The second nequalty follows from Lemma 3.3. Therefore f we sum the rght hand sde over all j e, the contrbuton from edges n that level wll be 3(2α) je C j e. Takng the contrbuton of both (1) and (2) together, the total cost of Γ s at most j=k j= j= 1 α j Cj + j=1 3(2α) j C j We can fnd a Stener Tree of cost at most ρ s tmes the above cost, so the lemma follows. Theorem 4.1 The Herarchy algorthm s a constant-approxmaton for the sngle-snk buy-atbulk problem. Proof: By Lemmas 4.2 and 4.3, the total expected cost of our soluton, E[N + P + T ], s bounded by 2(2T F + 2P I ). Usng Lemmas 4.4 and 4.6, we conclude that the expected cost of our soluton s bounded by the followng: 4 j=k ρ s j= j= 1 α j Cj + ρ s j=1 3(2α) j Cj j= + 2ρ f α j Cj j=1 By reversng orders of summaton, we can bound ths by: ( ρs 4 1 α + 6αρ s (1 2α) + 2ρ ) f C 1 α Ths s our approxmaton relatve to the near-optmum structured soluton. Usng Theorem 3.2 allows us to bound our overall approxmaton rato by: ( ) ( 4 ρs + 2ρ f 2α 2 1 α + 6αρ ) s 1 2α The best known approxmaton rato for Stener trees s ρ s = 1.55 due to [20], and that for faclty locaton s ρ f = 1.52 due to [18]. Settng α = 1/3 the above reduces to 54ρ f + 135ρ s < 292 approxmaton for the ncremental cost model. 11

13 5 Improved Approxmaton Algorthm for Access Network Desgn Andrews and Zhang [2] consder the case c = 1/2 and show that the optmal soluton can be converted wth a constant factor loss nto a layered soluton of shortest path forests. They show that there exsts a near-optmal (wthn a constant multpler on the cost) soluton whch s a tree satsfyng the followng propertes: 1. Each demand s routed through ppes of consecutve types,.e. types 1, 2,..., κ. (κ k). 2. For all ppe types k, any ppe of that type has at least u k /2 = σ k 2δ k through t. amount of demand flowng Ths means that for Access Network Desgn, the optmal soluton can be converted to a layered soluton usng shortest path forests of ncreasng ppe types. We can mprove the analyss of the above algorthm for Access Network Desgn. As shown n [12], for the Access Network Desgn we have a layered shortest path forest soluton wth a reducton n cost at each layer. We can prove the followng theorem: Theorem 5.1 There exsts a soluton to the Access Network Desgn problem n whch we only use ppe types satsfyng the condton φ = δ +1 δ α, and n whch any ppe of type has at least u /2 amount of demand flowng through t. The fxed and ncremental costs of ths soluton are each wthn 1 α of the orgnal optmum whch used all ppe types and whch had at least u k/2 demand n any ppe of type k. Proof: Note that snce we are usng ppes of larger types n ncreasng layers, the ncremental cost δ per unt of traffc keeps decreasng. In fact, we can make sure that δ goes down by a constant fracton α < 1 wth a 1 α ncrease n cost. The way we do ths s the followng: Consder ppes of ncreasng types startng at type 1. Let φ = δ +1 δ. Let k be the largest number such that k =1 φ α. We remove all ppe types 2,..., k + 1 and use only ppe of type 1 nstead of all these ppes. We next consder ppes startng at type k + 2 and repeat ths flterng process. When the above s completed, we are left wth a set of ppe types satsfyng the followng propertes. For consecutve ppe types and + 1, δ +1 δ α. Fnally, note that n ths process, f a ppe of type j s replaced by a ppe of type, t must be the case that φ < φ j, and δ < 1 α δ j, so that the cost of usng ppe s at most 1 α tmes the cost of usng ppe j. Recall φ = δ +1 δ. From above, we can assume wth a loss of 1 α n the approxmaton rato that all φ α < 1. Our algorthm wll lay ppes n ncreasng order of types. Let S denote the demand ponts at stage. We mantan the nvarant that every demand pont has at least u /6 demand. We solve the load balanced faclty locaton nstance on S wth lower bound u +1 (except on the snk s). We route the demands to the open facltes usng ppes of type. For every open faclty, we choose one of the demand ponts sendng demand to t at random n proporton to ts demand, and route all the demand to ths pont usng ppes of type + 1. Let S +1 be the fnal set of demand ponts to where we route the demands. Note that every demand pont has at least u +1 /6 demand. Let P I be the routng cost at stage, and let P F of the nvarant on the demands. be the fxed cost. Note that P F 6P I because We defne C to be the total ncremental cost ncurred by the optmal soluton usng ppes of type. Note that the total cost of the optmal soluton s C C. 12

14 Lemma 5.1 E[P I ] 2ρ f (1 + α)( j= 1 j=1 α j 1 C ). Proof: The routng cost that the optmum soluton pays n routng the orgnal demand ponts tll stage usng ppes of type s at most j= 1 j=1 α j 1 C. Ths follows from [12] and from the analyss n Secton 4. Ths s an nstance of the load balanced faclty locaton problem, and we apply Theorem 3.1. It s now easy to see the followng. Lemma 5.2 E[ (P I + P F )] 14ρ f 1+α 1 α C. Note that we lost a factor of 1 α up front n the routng cost because of scalng the ppe types. Our approxmaton rato s therefore 1 α 14ρ f 1+α 1 α. Settng α = 1/3 and ρ f = 1.52 ths rato s less than 128. Theorem 5.2 We have a randomzed 128 approxmaton for Access Network Desgn. Ths approxmaton factor has been subsequently mproved n [15]. 6 Load Balanced Faclty Locaton Recall the defnton of faclty locaton and load balanced faclty locaton from Secton 3.1. The load balanced problem dffers from standard faclty locaton [22] n that we must route at least L unts of demand to each open faclty. Load Balanced Faclty Locaton has drect applcatons; consder a franchse whch must open stores to mnmze the average dstance from customer to store, but whch must also guarantee a mnmum number of customers to each store so the ndvdual stores reman proftable. We present a constant approxmaton to ths problem, losng a constant factor compared to the lower bound on demand. We can wrte an nteger program for ths problem. Mnmze d j c j x j + j x j 1 j x j y, j j d jx j L y x j, y {0, 1}, j Clearly, the general verson of ths problem s NP-hard, as t reduces to classcal faclty locaton when the lower bounds are set to zero. In fact, ths problem s NP-hard even f all faclty costs are zero, all lower bounds are equal and all demands are unt. φ y Theorem 6.1 Suppose we are gven a load balanced faclty locaton nstance wth lower bound L on all faclty locatons, and faclty costs beng zero. Decdng f a feasble soluton of cost at most C exsts s NP-hard. Proof: We reduce the decson verson of the unweghted set cover problem to an nstance of ths problem as follows. The sets are the facltes. The elements are the demand ponts wth unt demand. Suppose there are n elements. In the faclty locaton nstance, we add edges of cost one between every element and all the sets t belongs to. 13

15 Suppose we have to decde f a cover wth s > 1 sets exsts. We add sn demand ponts wth unt demand, and connect them to all the sets (or facltes) wth edges of length one. We set the lower bound on the facltes to be n + 1. We now ask f there s a feasble soluton of cost no more than ns + n + s. Note that f there exsts a set cover of sze s, then there exsts a soluton of cost ns + n + s. The reverse also holds, and therefore ths completes the reducton. Defnton 6.1 An approxmaton algorthm for load balanced faclty locaton s a (α, β) approxmaton for some α 1 and β 1 f the cost of the soluton s wthn α tmes the optmal cost and faclty, f opened, serves at least L β demand. Let us denote by ρ f the best known approxmaton rato for classcal faclty locaton, whch s ρ f = 1.52 due to [18]. We present a (2ρ f, 1/3) approxmaton to ths problem. The same result was ndependently obtaned by [16]. Unlke classcal faclty locaton [22], the lower bound makes t hard to round the lnear relaxaton drectly. Ths arses from the fact that the flterng steps of Ln and Vtter n [17] do not work. Thus fractonal solutons cannot be rounded by prevous approaches. The Algorthm: The algorthm proceeds n two basc steps and uses an approxmaton algorthm for the faclty locaton problem. We note that the approxmaton guarantee holds relatve to the LP relaxaton as well (albet wth more techncal detals that we omt, snce t s not the man focus of the paper). Load Balanced Faclty Locaton Algorthm () Transformaton: For faclty, add the cheapest way to route exactly L unts of demand to to the faclty cost φ. To do ths, consder demands n ncreasng order of dstance from, and route these demands to untl exactly L unts have been routed. The routng cost of ths process s added to φ. (): Faclty Locaton: Next solve regular faclty locaton wth these faclty costs usng the ρ f -approxmaton algorthm. () Roundng to Remove Facltes: Now consder the open facltes n arbtrary order. Consder any open faclty that serves less than L /3 amount of demand. Close the faclty and route the demands t serves to ther closest open facltes. Lemma 6.1 Consder any feasble soluton to the load balanced faclty locaton problem of cost C. After the transformaton n Step (), ths yelds a feasble nstance of the regular faclty locaton problem of cost at most 2C. Proof: Consder any faclty opened by the load balanced soluton. Snce ths soluton s routng at least L amount of demand to any open faclty, the faclty cost we assgn n the new problem s at most the routng cost of the demand connected to that faclty. Thus the total addtonal faclty cost s at most C. Therefore the total cost n the soluton we compute s bounded n terms of the cost of the orgnal soluton to wthn a factor of 2ρ f. Also note that faclty locaton guarantees that each demand pont goes to the closest open faclty. We have to show that removng a faclty does not ncrease the total faclty plus routng cost of the soluton. For ths, we show a feasble way to route the demands t serves so that the cost does not ncrease. Lemma 6.2 Removng a faclty servng less than L 3 of our soluton. amount of demand cannot ncrease the cost 14

16 Proof: Suppose we are closng faclty. Consder the closest demand pont j whch does not send demand to ths faclty. Suppose c j = D. If j s beng served by, c j < D, as each demand pont goes to the closest open faclty. Note that at least 2L /3 unts of demand are at dstance D or greater. Therefore, f 2L 3 D. When we close the faclty, we can afford to use f towards re-routng the demand t serves. We send the demand to, the faclty servng j. The extra cost for dong ths s at most the cost of takng the demand from to j and from there to. Ths dstance s at most 2D by the metrc property, and the demand s at most L 3, and so the total re-routng cost s at most 2L 3 D. The above can be summarzed n the followng theorem: Theorem 6.2 The load balanced faclty locaton problem has a (2ρ f, 1/3) approxmaton where each demand s served by ts closest open faclty. 7 Concluson In ths paper, we presented the frst constant factor approxmaton for the sngle-snk buy-at-bulk network desgn problem. We conclude by surveyng the results on ths and related problems that have appeared snce the publcaton of the prelmnary verson of ths paper [13]. Frst, the algorthm tself has been sgnfcantly mproved and smplfed. Gupta, Kumar, and Roughgarden [15] obtan a 72.8 approxmaton by combnng the Stener and shortest path stages nto a rent-or-buy stage, and usng a novel analyss. Ths s the current best known approxmaton guarantee. For the sngle-snk case, Goel and Estrn [11] consder smultaneous (oblvous) approxmaton over all concave functons, and obtan a O(log n) approxmaton. For the multple snk (source-snk pars) verson of ths problem, as mentoned earler, the best known approxmaton rato of O(log n) follows drectly from tree embeddngs [3, 9]. For ths verson, Andrews [1] has shown a Ω((log n) 1 4 ) hardness of approxmaton assumng NP Dtme(n logo(1) n ). The non-unform verson of the problem assumes dfferent cable types are avalable on dfferent edges. As mentoned earler, the best known approxmaton rato [19] for the sngle-snk verson s O(log n). Chuzhoy et al. [8] show that the sngle snk verson s hard to approxmate wthn Ω(log log n), under smlar hardness assumptons as the unform case. Charkar and Karagozova [6] consder the non-unform verson n the presence of multple source-snk pars. The best result for ths case s a polylogarthmc approxmaton rato, and s acheved by Chekur et al. [7]. Acknowledgments We thank Matthew Andrews, Chandra Chekur, and Serge Plotkn for several helpful dscussons. Ths work was done whle the authors were at Stanford Unversty. Sudpto Guha was supported by an IBM Research Fellowshp, NSF Grant IIS and NSF Award CCR , wth matchng funds from IBM, Mtsubsh, Schlumberger Foundaton, Shell Foundaton, and Xerox Corporaton. Adam Meyerson and Kamesh Munagala were supported by ARO grant DAAG and ONR grant N References [1] M. Andrews. Hardness of buy-at-bulk network desgn. In Proceedngs of the 45th Annual Symposum on Foundatons of Computer Scence, pages ,

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