v e v 1 C 2 b) Completely assigned T v a) Partially assigned Tv e T v 2 p k

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1 Approximation Algorithms for a Capacitated Network Design Problem R. Hassin 1? and R. Rai 2?? and F. S. Salman 3??? 1 Department of Statistics and Operations Research, Tel-Ai Uniersity, Tel Ai 69978, Israel. hassin@math.ta.ac.il 2 GSIA, Carnegie Mellon Uniersity, Pittsbrgh, PA rai@cm.ed 3 GSIA, Carnegie Mellon Uniersity, Pittsbrgh, PA fs2c@andrew.cm.ed Abstract. We stdy a network loading problem with applications in local access network design. Gien a network, the problem is to rote ow from seeral sorces to a sink and to install capacity on the edges to spport ows at minimm cost. Capacity can be prchased only in mltiples of a xed qantity. All the ow from a sorce mst be roted in a single path to the sink. This NP-hard problem generalizes the Steiner tree problem and also more eectiely models the applications traditionally formlated as capacitated tree problems. We present an approximation algorithm with performance ratio ( ST +2) where ST is the performance ratio of any approximation algorithm for minimm Steiner tree. When all sorces hae the same demand ale, the ratio improes to ( ST + 1) and in particlar, to 2 when all nodes in the graph are sorces. 1 Introdction We consider a single-sink mltiple-sorce roting and capacity installation problem where capacity can be prchased in mltiples of a xed qantity. In telecommnication network design this corresponds to installing transmission facilities sch as ber-optic cables on the edges of a network, and in transportation networks this applies to assigning ehicles of xed capacity to rotes. Topological design of commnication networks is sally carried in stages de to the complexity of the problem. One of the fndamental stages is the design of a local access network which links the sers to a switching center. The problem we stdy models this stage of the planning process. Problem statement. We are gien an nderlying ndirected graph G = (V; E); jv j = n. A sbset S of nodes is specied as sorces of trac and a single sink t is specied. Each sorce node s i 2 S has a positie integer-aled demand? This work was done when this athor isited GSIA, Carnegie Mellon Uniersity.?? Spported by an NSF CAREER grant CCR ??? Spported by an IBM Corporate Fellowship.

2 dem i. All the trac of each sorce mst be roted to t ia a single path, that is ow cannot be bifrcated. The edges of G hae lengths ` : E! < +. Withot loss of generality, we may assme that for eery pair of nodes ; w, we can se the shortest-path distance dist(; w) as the length of the edge between and w; Therefore, we take the metric completion of the gien graph and assme all edges from the complete graph are aailable. Capacity mst be installed on the edges of the network by prchasing one or more copies of a facility, which we refer to as the \cable" based on the telecommnication application. The cable has per nit length cost c and capacity. Withot loss of generality we can assme c = 1. The problem is to specify for each sorce s i, a path to t to rote demand dem i sch that cables installed on each edge of the network proide scient capacity for the ow on the edge, and total cost of cables installed is minimized. Notice that we allow paths from dierent sorces to share the capacity on the installed cables, the only restriction being that the capacity installed on an edge is at least as mch as the total demand roted throgh this edge. The problem is NP-hard since the problem with cable capacity large enogh to hold all of the demand is eqialent to a Steiner tree problem with the sorces and the sink as the terminal nodes. Preios Work. This problem has been stdied in the literatre as the network loading problem, together with its ariations sch as the mlticommodity and mltiple facility cases. For a srey on exact soltion methods see the chapter on mlticommodity capacitated network design by Gendron, Crainic and Frangioni in [SS99]. In spite of the recent comptational progress, the size of the instances that can be soled to optimality in reasonable time is still far from the size of real-life instances. In this paper we focs on obtaining approximation algorithms. A constant factor approximation for this problem was obtained by Salman et al. in [SCR+97] by applying the method of Mansor and Peleg [MP 94] to the case of single sink, and single cable type. The main algorithm of Mansor and Peleg applies to the mltiple-sorce mltiple-sink single cable problem with approximation ratio O(log n) in an n-node graph. By sing a Light Approximate Shortest-Path Tree (LAST) [KRY 93] instead of a more general-prpose spanner in this algorithm, Salman et al. obtained a 7-approximation algorithm for the single-sink ersion. When all the nodes in the inpt network except the sink node are sorce nodes, the approximation ratio in [SCR+97] redces to (2 p 2 + 2). Another constant factor approximation algorithm for this problem also follows from the work of Andrews and Zhang [AZ98] who gae an O(k 2 )-approximation algorithm for the single sink problem with k cable types, bt the reslting constant factor is rather high. Reslts. In this paper, we improe the approximation ratio to ( ST + 2) by roting throgh a network that is bilt on an approximate Steiner tree, with performance ratio ST. The idea is to tilize the Steiner tree when demand is low compared to the cable capacity and when demand accmlates to a ale close to the cable capacity, it is sent directly to the sink. For the special case

3 when demand of each sorce is niform, the approximation ratio improes to ( ST + 1). When all the nodes in the inpt network except the sink node are sorce nodes, the approximation ratio redces to 3 with non-niform demands, and to 2, for niform demands. Or stdy was also motiated by obtaining better approximation algorithms for the capacitated MST problem [Pap78,AG88,KB83,CL83,S83]: Gien an ndirected edge-weighted graph with a root node and a positie integer, the problem is to nd the minimm weight tree sch that eery sbtree hanging o the root node has at most nodes in it. This problem has been cited [KR98,AG88] to model the local access network design problem when eery non-sink node is reqired to rote a single nit of demand to the sink ia cables each of capacity at most. The reqirement that eery demand has to send its nit ow ia a single path is modeled as reqiring a tree as the soltion. Howeer, if roting these demands at nodes is not a concern, we can still enforce the nonbifrcating reqirement for the demands withot reqiring that the soltion be a tree. This reformlation leads exactly to or single cable problem in the niform case with all nodes being sorces. Or 2-approximation algorithm for this problem is then a better soltion than the best-known 3-approximation [AG88] for the corresponding capacitated MST formlation. In the nonniform demand case, or ( ST +2)-approximation is better than the best known 4-approximation presented in [AG88] in addition to handling the Steiner ersion that does not reqire all non-sink nodes be sorce nodes. In the next two sections, we present the algorithms for the case of niform and non-niform demands, respectiely. We close with an extension of the local access design problem. 2 Uniform Demand We rst present an approximation algorithm for the case when eery sorce has the same demand. Withot loss of generality, we assme demand eqals one for each sorce. We can otline the algorithm as follows. First we constrct an approximate Steiner tree with terminal set S[ftg and cost dist(e) on each edge e in polynomial time. Let T be the approximate Steiner tree with worst-case ratio ST 1. Let the tree T be rooted at the sink node t. Next, we identify sbtrees of T sch that total demand in a sbtree eqals the cable capacity. We then rote the total demand within a sbtree directly to the sink from the node of the sbtree closest to the sink. The sbtrees collected by the algorithm may contain common nodes bt hae disjoint sorce sets. For a formal statement of the algorithm, we need the following denitions. Let the leel of a node be the nmber of tree edges on its path to the root. The parent of a node is the node adjacent to it on the path from to t. For each node, let T denote the sbtree of T rooted at and D(T ) denote the total 1 The MST is a 2-approximate soltion. Better approximation ratios are known, e.g., a 1.55-approximation was gien recently in [RZ00].

4 nprocessed P demand in T. Let R be the set of nprocessed sorce nodes. Then, D(T ) = s i2r\t dem i = jr \ T j. The Algorithm Uniform below otpts a roting for the demand from each sorce to the sink, and the nmber of cables that are installed to spport the roting. Algorithm Uniform: Initialize: R = S Main step: Pick a node sch that D(T ) and leel of is maximm. If = t or D(T t ) <, then go to the nal step. Pick a node, say w in R \ T sch that dist(w; t) is minimm, as a \hb" node. Let C = fwg. Collect sorce nodes in C (Details gien below). Add edge (w; t) to the network and install one copy of the cable on (w; t). Rote demand of each sorce in C to the hb node w ia the niqe paths in T Rote demand of C at the hb directly to the sink on (w; t). Remoe C from R, and set C = ;. If R is not empty, repeat the main step. If R is empty, go to the nal step. Final step: If R 6= ;, then rote all the demand in R to t ia their path in T. For all edges e of T, Cancel the maximal possible amont of ow of eqal ale in opposite directions sch that total ow will not exceed. Install one copy of cable on the edges of T which hae positie ow. Collect sorce nodes: Add to C, if 2 R. Let 1 ; : : : ; k be the children of. If w 6=, then Let p be the child of sch that the hb node w is in T p. Add T p \ R to C. While jcj <, Pick an nprocessed child of, say i. If D(T i ) + jcj, then Add T i \ R to C. Else, (T i is collected partially) Scan T i depth-rst. Add sorces in R \ T i to C ntil jcj =. Retrn C. Lemma 1. The algorithm rotes demand sch that ow on any edge of the tree T is at most the cable capacity. Proof. Consider an edge e of T. Let be the incident node on e with higher leel (see Figre 1). Flow on e is determined by the total ow coming ot of T and

5 t 1 e p k w T Fig. 1. Sbtree T and its children. going into T. Or proof is based on these two claims: Claim 1: Total ow going ot of T is at most 1. Claim 2: Total ow coming into T is at most 1. To proe claims 1 and 2, we consider two cases based on how the sorces in T are assigned to hb nodes by the algorithm. A partially assigned sbtree has at least one of its sorce nodes collected in a set C and has at least one sorce node not in C. t C1 e C T 2 C 1 e T t C 2 a) Partially assigned T b) Completely assigned T Fig. 2. Examples of partially and completely assigned sbtrees. Sppose T is partially assigned (see Figre 2). The rst time ow goes ot of T, a sbtree T with at a smaller leel than is being processed by the algorithm. De to the sbtree selection rle, we can conclde that T has remaining demand strictly less than. Therefore, total otow from T will be at most 1. Hence, Claim 1 holds in this case.

6 The reason Claim 2 holds is as follows. When there exists an inow into T, ow is accmlated at a hb node in T. Since the algorithm accmlates a ow of exactly at any hb node, a ow of at most 1 will go into T. The algorithm rst picks a sbtree and a hb node in it, and collects demand starting with the sbtrees of T. Therefore, the algorithm will not collect sorces ot of T, nless all the sorces in T hae already been collected. This implies that once ow enters T, none of the nodes in T will become a hb node again and hence ow will not enter T again. Now let s assme that T is not partially assigned. Then all the sorces in T are collected in the same set by the algorithm. If these sorces are roted to a hb node ot of the sbtree, then otow is at most 1. If the sorces are roted to a hb node in the sbtree, then inow is at most 1. Inow or otow occrs only once. Ths, Claims 1 and 2 hold in this case, too. For any edge of T, ow in one direction does not exceed, by Claims 1 and 2. When there exists ow in both directions in an edge with total ale greater than, we cancel ow of eqal ale in opposite directions sch that total ow will not exceed. Cancelling ow will lead to reassigning some of the sorce nodes to hbs. See Figre 3 for an example. C 1 : C 2 : t C 1 : C : t f in(e)=7 01 f 01 in(e)=2 01 e e f ot(e)=5 f ot(e)= w w w2 01 w a) On edge e, sm of flow in both b) Sorces are reassigned to hbs after directions exceeds, where =10. flow of ale 5 is cancelled on edge e. Fig. 3. An example of cancelling ow and reassigning sorces to hb nodes. Here w 1 and w 2 are hb nodes chosen in the order of their indices. Theorem 1. There is a (1 + ST )-approximation algorithm for the single-sink capacity installation problem with a single cable type and niform demand. Proof. Consider Algorithm Uniform. Let C OP T be the cost of an optimal soltion and C HEUR be the cost of a soltion otpt by the algorithm. Let C ST denote the cost of the cables installed on the edges of the approximate Steiner tree

7 T. Let C DR be the cost of cables installed on the direct edges added by the algorithm. By Lemma 1, at most one copy of cable is scient to accomodate ow on the edges of the approximate Steiner tree T. The cost of a Steiner tree with terminal set S [ ftg is a lower bond on the optimal cost becase we mst connect the nodes in S to t and install at least one copy of the cable on each connecting edge. Therefore, C ST ST C OP T. For a sorce set C k collected at iteration k, since jc k j =, the algorithm installs one copy of the cable on the shortest direct edge from the sbtree T, dem i P which contains C k, to t. The term s i2s dist(s i ; t) is a lower bond on C OP T, since dem i mst be roted a distance of at least dist(s i ; t) and be charged at least at the rate 1= per nit length. (In the niform demand case, dem i = 1 for all i.) Since sorce sets collected by the algorithm are disjoint, P dem i k P P dist(s i;t) is a lower bond on C OP T, P dist(s i ; t) = k as well. As demand of a set C k is sent ia the sorce in C k that is closest to t (the hb node w k ), we get X Ps dist(w k ; t) = min dist(s i ; t) i2c k dist(s i ; t) dist(s i ; t) = : (1) P 1 Ths, we nally hae C DR = X k dist(w k ; t) X k X dist(s i ; t) C OP T : (2) Therefore, C HEUR = C ST + C DR (1 + ST )C OP T : 3 Non-niform Demand When sorce nodes hae arbitrary demand, dem i for sorce s i, it is no longer possible to collect sorces with total demand exactly eqal to the capacity. If we were allowed to split the (integral) demand for any sorce into single integral nits each of which can be roted in separate paths to the sink, notice that the algorithm of the preios section can be sed by expanding each sorce s i to dem i sorces connected by zero-length edges in the tree. Howeer, in the more general case, all the ow of a sorce mst se the same path to the sink. In this case, we modify Algorithm Uniform so that we send demand directly to the sink when it accmlates to an amont between =2 and. To garantee that we don't exceed while collecting demand, we send all sorces with demand at least =2 directly at the beginning of the algorithm. For a sorce set C, let dem(c) be the total demand of sorces in C. Recall that D(C) is the total remaining (nprocessed) demand of C, as dened for the niform demand case. The modied algorithm, which we call Algorithm Non-niform, is as follows.

8 Algorithm Non-niform: Initialize: R = S. Preprocessing: (send large demands directly) For all sorces s i sch that dem i =2, Rote the demand on (s i ; t). Install d dem i e copies of cable on (s i; t). Remoe s i from R. Main step: Pick a node sch that D(T ) =2 and leel of is maximm. If = t, or D(T t ) < =2, then go to the nal step. Pick a node, say w in R \ T sch that dist(w; t) is minimm, as a \hb" node. Let C = fwg. Collect sorce nodes in C (Details gien below). Add edge (w; t) to the network and install one copy of the cable on (w; t). Rote demand of each sorce in C to the hb node ia the niqe path in T. Rote demand of C at the hb directly to the sink on (w; t). Remoe C from R and set C = ;. If R is not empty, repeat the main step. If R is empty, go to the nal step. Final step: If R 6= ;, then rote all the demand in R to t ia the niqe paths in T. Install one copy of cable on the edges of T which hae positie ow. Collect sorce nodes: Add to C, if 2 R. Let 1 ; : : : ; k be the children of. If w 6=, then Let p be the child of sch that the hb node w is in T p. Add T p \ R to C. While dem(c) < =2, Pick an nprocessed child of, say i. Add T i \ R to C. Retrn C. Lemma 2. The algorithm rotes demand sch that: 1) ow on any edge of the tree T is at most, and 2) ow on a direct edge added by the algorithm is at least =2 and at most. Proof. The proof is simpler compared to the niform-demand case becase the algorithm does not assign any sbtree partially. Consider an edge e of T. Let be incident on e sch that e is not in T. Since all the sorces in T are collected in the same set by the algorithm, demand of these sorces is roted to a hb node either ot of the sbtree, or in the sbtree, bt not both. Ths, ow exists only in one direction. If the demand of sorces is roted to a hb node ot of

9 T, then otow is at most 1. If the demand is roted to a hb node in the sbtree, then inow is at most 1. Ths, for any edge of T, ow does not exceed. De to the sbtree selection rle in the algorithm, if a sbtree T is selected, then all the sbtrees rooted at its children hae remaining demand strictly less than =2. Therefore, the rst time dem(c) exceeds =2, it will be at most so that total ow on the direct edges added by the algorithm is in the range [=2; ]. Theorem 2. There is a (2 + ST )-approximation algorithm for the single-sink edge installation problem with a single cable type and non-niform demand. Proof. We se the same denitions of C OP T, C HEUR, C DR and C ST as in the proof of Theorem 1. By Lemma 2, at most one copy of the cable is scient to accommodate ow on the edges of the approximate Steiner tree T. Therefore, C ST ST C OP T. For a sorce set C k collected at iteration k, the algorithm installs one copy of the cable on the shortest direct edge from the sbtree T, which encloses C k, to t. By Lemma 2, at most one copy of cable is scient to accommodate ow on the direct edges from hb nodes to t and dem(c k ) =2. The term P dem i s i2s dist(s i; t) is a lower bond on C OP T as in the niform demand case. Since sorce sets collected by the algorithm hae disjoint sorces and demand from a set C k is sent ia the sorce in C k that is closest to t (the hb node w k ), C OP T X k X X dem i dist(s i; t) k X dem i ( min dist(s i ; t)): (3) P Since dem i and min 2 X dist(s i ; t) = dist(w k ; t), we hae C OP T X 1 2 dist(w k; t) = 1 2 C DR: (4) k Therefore, C HEUR = C ST + C DR (2 + ST )C OP T : 4 Extensions Or methods apply to the following extension of the local access network design problem: Instead of specifying a single sink node, any node in the graph can be sed as a node that sinks nits of demand at a cost of f. A node is allowed to sink more than nits of demand by paying d dem e f cost to sink dem nits of ow. The problem is to open scient nmber of sinks and rote all the demands to these sinks at minimm cable pls sink opening costs. To model this extension, we extend the metric in two steps: 1) create a new sink node t with edges to eery ertex of cost f, 2) take the metric completion of this agmented network. Notice that the second step may decrease some of the costs on the edges incident on the new sink t (e.g., if f i + dist(j; i) < f j,

10 then the cost of the edge (j; t) can be redced from f j to f i + dist(j; i)), or between any pair of original nodes (e.g., if dist(i; j) > f i + f j, then we may replace the former by the latter). Bearing this in mind, it is not hard to see that any soltion in the new graph to the single cable problem with t as the sink and with the modied costs can be conerted to a soltion to the original problem of the same cost. Ths, or algorithms in the preios sections apply to gie the same performance garantees. References [AG88] K. Altinkemer and B. Gaish, \Heristics with constant error garantees for the design of tree networks," Management Science, 34, (1988) 331{341 [AZ98] M. Andrews and L. Zhang, \The access network design problem," In Proc. of the 39th Ann. IEEE Symp. on Fondations of Compter Science, (1998) 42{49 [CL83] K. M. Chandy and T. Lo, \The capacitated minimm tree," Networks, 3, (1973) 173{182 [KR98] Kawatra, R. and D. L. Bricker, "A mltiperiod planning model for the capacitated minimal spanning tree problem", to appear in Eropean Jornal of Operational Research (1998) [KB83] A. Kershenbam and R. Boorstyn, \Centralized teleprocessing network design," Networks, 13, (1983) 279{293 [KRY 93] S. Khller, B. Raghaachari and N. E. Yong, \Balancing minimm spanning and shortest path trees," Algorithmica, 14, (1993) 305{322 [MP 94] Y. Mansor and D. Peleg, \An approximation algorithm for minimm-cost network design," The Weizman Institte of Science, Rehoot, Israel, Tech. Report CS94-22, 1994; Also presented at the DIMACS workshop on Robst Commnication Networks, [Pap78] C. H. Papadimitrio, \The complexity of the capacitated tree problem," Networks, 8, (1978) 217{230 [RZ00] G. Robins and A. Zelikosky, \Improed steiner tree approximation in graphs", Proc. of the 10th Ann. ACM-SIAM Symp. on Discrete Algorithms, (2000) 770{779 [SCR+97] F.S. Salman, J. Cheriyan, R. Rai and S. Sbramanian, \By-at-blk network design: Approximating the single-sink edge installation problem," Proc. of the 8th Ann. ACM-SIAM Symposim on Discrete Algorithms, (1997) 619{628 [S83] R. L. Sharma, \Design of an economical mltidrop network topology with capacity constraints," IEEE Trans. Comm., 31, (1983) [SS99] B. Sanso and P. Soriano, Editors, \Telecommnications Network Planning," Klwer Academic Pblishers, 1999.