G. Robins and A. Zelikovsky. Improved Steiner Tree Approximation in Graphs. In Proc. 10th Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 770-779, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sum of edge lengths of a subgraph U of G by l(U) e2E(U) l(e) The time for computing an r approximate optimum Steiner tree will be denoted by . Throughout the paper, r 1:55 and 2 (0; 1] will be constant parameters describing the guaranteed ratio of our minimum Steiner tree approximation [18] and determining the target approximation of the fractional optimization problem, respectively. MST l (S i ) and MST l (S i ) are used to denote a minimum Steiner tree and an approximate minimum Steiner tree for S i with respect to l (l is usually omitted) we assume l( MST(S i ) r l(MST(S ....

G. Robins, A. Zelikovsky, Improved Steiner tree approximation in graphs, Proc. of the 11th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA 2000), 770{ 779, 2000.

.... and Multicasting in Adversarial Systems: Routing and Admission Control (Preliminary Technical Report) Baruch Awerbuch # Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218, USA baruch cs.jhu.edu Andre Brinkmann Heinz Nixdorf Institute and Department of Electrical Engineering University of Paderborn 33102 Paderborn, Germany brinkman hni.upb.de Christian Scheideler Department of Computer Science Johns Hopkins ....

.... Street Baltimore, MD 21218, USA baruch cs.jhu.edu Andre Brinkmann Heinz Nixdorf Institute and Department of Electrical Engineering University of Paderborn 33102 Paderborn, Germany brinkman hni.upb.de Christian Scheideler Department of Computer Science Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218, USA scheideler cs.jhu.edu Johns Hopkins University, March 01 2002 Abstract In this paper we consider the problem of routing packets in dynamically changing networks, concentrating on two different modes: anycasting and multicasting. In anycasting, a ....

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proc. of the 11th ACM Symp. on Discrete Algorithms (SODA), pages 770--779, 2000.

....with economies of scale. Our Results Our main results are the following. 1. We give a randomized approximation algorithm for CFL with a performance bound of 2 #ST , using a #ST approximation algorithm for the Steiner tree problem; the currently smallest available value for #ST is 1. 55 [23]. This simple, intuitive and easily analyzed algorithm improves over the previously best known guarantee of 3 #ST , due to Swamy and Kumar [25] 2. We resolve the main open question posed in [13] by giving a 5.55 approximation algorithm for virtual private network design. Previously, ....

....2. CONNECTED FACILITY LOCATION In this section, we present an intuitive and easy to implement randomized approximation algorithm for CFL with performance guarantee 2 #ST , using a #ST approximation algorithm for the Steiner tree problem. With the Steiner tree algorithm of Robins and Zelikovsky [23], we obtain a 3.55 approximation, improving upon the primal dual 4.55 approximation algorithm of Swamy and Kumar [25] We recall from Section 1 that in an instance of CFL, we are given an undirected graph G = V, E) with non negative edge costs ce , V of demands, and a parameter M 1. The ....

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Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770--779, 2000.

....an approximation ratio of 2 (i.e. the tree that it outputs can have no more than 2 times as many edges as the optimum) A distributed version of this algorithm is discussed in [2] The best known approximation algorithm for the minimum Steiner tree problem has an approximation ratio of about 1. 55 [30]. Finally, we mention here in passing that there is another sense in which the phrase datacentric networking has been used [8] namely to describe an approach to ubiquitous computing in which human users are identified not with static computing devices but with their personalized services and ....

G. Robins, A. Zelikovsky, "Improved Steiner Tree Approximation in Graphs," Proc. of ACM/SIAM Simposium on Discrete Algorithms, pp. 770-779, 2000.

....at least since 1968 [6, p. 24] that the performance ratio of the minimum spanning tree heuristic is 2. During the last ten years, several authors published algorithms with decreasing performance ratios [18, 3, 12, 19, 11, 9] The best value known today is 1:550 and due to Robins and Zelikovsky [15]. For more details on these approximation algorithms see [8] The PCP Theorem [1] and an approximation preserving reduction from vertex cover [4] imply that the performance ratio of a polynomial time approximation algorithm for the Steiner tree problem in graphs cannot get arbitrarily close to 1. ....

....quasi bipartite graphs. In these instances, the set V nR is stable, but the edges incident with a vertex in that set may have different lengths. Rajagopalan and Vazirani [13] gave a 3=2 approximation algorithm based on the primal dual method for quasi bipartite instances. Robins and Zelikovsky [15] showed that the popular 1 Steiner heuristic has a performance ratio of 3=2 in this case. Moreover, they showed that the performance ratio of their loss contracting algorithm is 1:279 in quasi bipartite instances. 2 Algorithm Greedy MSS Greedy MSS Greedy MSS Every Steiner tree can be split into ....

G. Robins, A. Zelikovsky, Improved Steiner tree approximation in graphs, In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2000, 770--779.

....the root. The parent of a node v is the node adjacent to it on the path from v to t. For each node v, let Tv denote the subtree of T rooted at v and D(Tv) denote the total The MST is a 2 approximate solution. Better approximation ratios are known, e.g. a 1. 55 approximation was given recently in [RZ00] unprocessed demand in Tv. Let R be the set of unprocessed source nodes. Then, D(Tv) 8cRn demi = I R N Tvl. The Algorithm Uniform below outputs a routing for the demand from each source to the sink, and the number of cables that are installed to support the routing. Algorithm Uniform: ....

G. Robins and A. Zelikovsky, "Improved steiner tree approximation in graphs", Proc. of the 10th Ann. ACM-SIAM Symp. on Discrete Algorithms, (2000) 770-779

....10099 Berlin, Germany, groepl, hougardy, nierhoff, proemel informatik.hu berlin.de. heuristic is 2. During the last ten years, several authors published algorithms with decreasing performance ratios [3, 9, 11, 12, 18, 19] The best value known today is 1. 550 and due to Robins and Zelikovsky [15]. For more details on these approximation algorithms see [8] The PCP Theorem [1] and an approximation preserving reduction from vertex cover [4] imply that the performance ratio of a polynomial time approximation algorithm for the Steiner tree problem in graphs cannot get arbitrarily close to 1. ....

....graphs. In these instances, the set V R is stable, but the edges incident with a vertex in that set may have different lengths. Rajagopalan and Vazirani [13] gave a 3 2 # approximation algorithm based on the primal dual method for quasi bipartite instances. Robins and Ze1 likovsky [15] showed that the popular 1 Steiner heuristic has a performance ratio of 3 2 in this case. Moreover, they showed that the performance ratio of their loss contracting algorithm is 1.279 in quasi bipartite instances. 2 Algorithm Greedy MSS Greedy MSS Greedy MSS Every Steiner tree can be split into ....

G. Robins, A. Zelikovsky, Improved Steiner tree approximation in graphs, In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2000, 770--779.

....it from the similar minimum spanning tree problem, for which there is a simple polynomial time algorithm. Most of the theoretical work on the Steiner tree problem concerns obtaining approximating algorithms. Currently the best approximation factor is 1. 55, obtained by Robins and Zelikovsky [16]. Arora [2] showed that an approximation can be achieved for every 0 when the underlying graph is Euclidean. On the other hand, unless P = NP , the Steiner tree problem in general graphs can not be approximated within a factor of 1 for some 0, see [4] 7] In this paper we focus on ....

G. Robins and A. Zelikovsky, Improved Steiner tree approximation in graphs, Proc. 11th ACMSIAM Symposium on Discrete Algorithms (2000), 770-779.

....an approximation ratio of 2 (i.e. the tree that it outputs can have no more than 2 times as many edges as the optimum) A distributed version of this algorithm is discussed in [2] The best known approximation algorithm for the minimum Steiner tree problem has an approximation ratio of about 1. 55 [30]. Finally, we mention here in passing that there is another sense in which the phrase data centric networking has been used [8] namely to describe an approach to ubiquitous computing in which human users are identified not with static computing devices but with their personalized services and ....

G. Robins, A. Zelikovsky, "Improved Steiner Tree Approximation in Graphs," Proc. of ACM/SIAM Simposium on Discrete Algorithms, pp. 770-779, 2000.

....tighter analysis and randomization. For source limited case where each threshold is (b in = #, b out = the arguments of Theorem 2.4 show that the optimal tree solution corresponds exactly to the optimal Steiner tree connecting the terminal set, and hence can be approximated to within 1. 55 [18]. 3 Unsplittable and Fractional Graph Solutions 3.1 The Symmetric Case In this section, we show that a solution to SymT is a constant factor approximation to SymG. In fact, we show something much stronger; i.e. a solution to SymT is within 2(1 1 k) of the optimal fractional solution to ....

Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770--779, 2000.

....possible using randomization. For the source limited case where each threshold is (b in = #, b out = the arguments of Theorem 2.4 10 show that the optimal tree solution corresponds exactly to the optimal Steiner tree connecting the terminal set, and hence can be approximated to within 1. 55 [19]. 3 Unsplittable and Fractional Graph Solutions 3.1 The Symmetric Case In this section, we show that a solution to SymT is a constant factor approximation to SymG. In fact, we show something stronger; i.e. a solution to SymT is within 2(1 1 k) of the optimal fractional solution to SymF . ....

Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770--779, 2000.

....and traveling purchaser problems. They provide fast combinatorial algorithms for the weighted versions of these problems achieving approximation ratios 5.5 and 3.55 respectively (3. 55 is slightly lower than their claim the reason being the recent improvements in minimum Steiner tree approximation [8]) For unweighted versions their best approximation ratios are 3 (tour cover) and 2 (tree cover) and they also show how to nd a 3 approximate tree cover in linear time. Finally, they give approximation preserving reductions to vertex cover and traveling salesman problem, showing that tree and ....

G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770-779, 2000.

....it from a similar minimal spanning tree problem for which there exists a simple polynomial time algorithm. Most of the theoretical work on Steiner tree problem is concentrated on obtaining approximating algorithms and currently the best approximating factor is 1. 55 due to Robins and Zelikovsky [8]. An approximation can be achieved for every 0 when the underlying graph is Euclidean, Arora [1] On the other hand, unless P = NP , the Steiner Tree Problem in general graphs can not be approximated within factor 1 for some suciently small 0, see [2] 3] In this paper we consider ....

G. Robins and A. Zelikovsky, Improved Steiner Tree Approximation in Graphs, Proc. 11th ACMSIAM Symposium on Discrete Algorithms (2000), 770-779. 13

....The parent of a node v is the node adjacent to it on the path from v to t. For each node v, let T v denote the subtree of T rooted at v and D(T v ) denote the total 1 The MST is a 2 approximate solution. Better approximation ratios are known, e.g. a 1. 55 approximation was given recently in [RZ00] unprocessed demand in T v . Let R be the set of unprocessed source nodes. Then, D(T v ) P s i 2R Tv dem i = jR T v j. The Algorithm Uniform below outputs a routing for the demand from each source to the sink, and the number of cables that are installed to support the routing. Algorithm ....

G. Robins and A. Zelikovsky, "Improved steiner tree approximation in graphs", Proc. of the 10th Ann. ACM-SIAM Symp. on Discrete Algorithms, (2000) 770--779

....length of this Steiner tree comes arbitrarily close to the length of a Steiner minimum tree. Therefore, good approximation algorithms to the k MSS problem yield also good approximation algorithms for the Steiner tree problem. All recent approximation algorithms for solving the Steiner tree problem [17, 19, 1, 20, 15, 9, 8, 16] are based on this approach. The approximation algorithm for k MSS uses a similar greedy strategy as Chvatal s algorithm for k set cover. However, the analysis needs some new idea. The main reason for this is that the connectedness of the subhypergraph as required in a solution to k MSS is ....

Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proc. Symposium on Discrete Algorithms, pages 770--779, 2000.

....and traveling purchaser problems. They provide fast combinatorial algorithms for the weighted versions of these problems achieving approximation ratios 5.5 and 3.55 respectively (3. 55 is slightly lower than their claim the reason being the recent improvements in minimum Steiner tree approximation [8]) For unweighted versions their best approximation ratios are 3 (tour cover) and 2 (tree cover) and they also show how to nd a 3 approximate tree cover in linear time. Finally, they give approximation preserving reductions to vertex cover and traveling salesman problem, showing that tree and ....

G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770-779, 2000.

....for computing a low cost (standard) Steiner tree. We denote by ff ST the approximation factor of the heuristic chosen. The simple Steiner tree heuristics (see, e.g. 19] achieve an approximation factor of 2. Alternatively, the current best value known for ff ST is by Robins and Zelikovsky [17] (following a long line of work) who suggested a heuristic with approximation factor of 1 ln 3 2 1:55. We use STEINER(n) to denote the time complexity of the Steiner tree heuristic used. The running times of our algorithms will be expressed in terms of STEINER(n) If we use the simple ....

....only experimental evidence for its performance. We present a heuristic for this problem and show that the cost of the Steiner tree that it generates is no more than e Delta ff ST times the cost of an optimal tree, where e is the basis of the natural log (e 2:7182) Plugging in ff ST = 1:55 [17], we get that the cost of the multicast tree generated by our algorithm is no more than 4:214 times the cost of an optimal multicast tree. We then turn our attention to the priority model. We present a heuristic for the minimum cost Steiner tree problem in this model and show that the cost of the ....

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G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. Proceedings of 11th ACM-SIAM SODA, 2000.

....the duals raised around proper sets are converted into a solution to (1) by picking t o and discarding y S s with S = 2 C. The 3 2 approximation guarantee follows by relating the cost of this solution to the cost of MST(T ) 1 1 Recently, using a different argument, Robins and Zelikovsky [22] proved that any Steiner tree satisfying condition (b) of Theorem 1 is within a factor of 3 2 of optimum. 8 C. Efficient implementation of the RV Phase algorithm Since our heuristic on general graphs uses RV Phase as a subroutine, we describe here an efficient implementation of it. Several ....

G. Robins and A. Zelikovsky. Improved Steiner Tree Approximation in Graphs, to appear in Proc. of ACM/SIAM Simposium on Discrete Algorithms (SODA'2000).

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G. Robins and A. Zelikovsky, Improved Steiner Tree Approximation in Graphs, Proc. of ACM/SIAM Symposium on Discrete Algorithms (SODA 2000), 770--779.

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th ACM-SIAM Annual Symposium on Discrete Algorithms, pages 770--779, 2000.

....al. 9] They give algorithms with approximation factors of 2e 5:44 and 16.86 for ZST and BST problems, respectively. The BST algorithm in [9] relies on an approximation algorithm for the Steiner tree problem in graphs. Using the currently best Steiner tree approximation of Robins and Zelikovsky [22] and Arora s PTAS for computing rectilinear Steiner trees [1, 2] the BST bounds in [9] can be updated to 16.11 for arbitrary metric spaces, and to 12.53 for the rectilinear plane (see Table 1) In this paper we introduce a new approach to these problems, based on zero skew stretching of ....

.... Runtime in this paper O(n 2 ) O(nlogn) O(n 2 ) O(nlogn) Table 1: Summary of results and comparison to results of Charikar et al. 9] Values marked with asterisks update those reported in [9] by taking in account the currently best Steiner tree approximation of Robins and Zelikovsky [22] and Arora s PTAS for computing rectilinear Steiner trees [1, 2] In Section 4 we give a Kruskal like algorithm that builds a rooted spanning tree T whose total delay does not exceed its length, and whose length is at most twice that of the optimal ZST. These two facts yield an approximation ....

ROBINS, G., AND ZELIKOVSKY, A. Improved Steiner tree approximation in graphs. In Proc. 11th ACMSIAM Symp. on Discrete Algorithms (2000), pp. 770--779. 18

....al. 9] They give algorithms with approximation factors of 2e 5:44 and 16.86 for ZST and BST problems, respectively. The BST algorithm in [9] relies on an approximation algorithm for the Steiner tree problem in graphs. Using the currently best Steiner tree approximation of Robins and Zelikovsky [22] and Arora s PTAS for computing rectilinear Steiner trees [1, 2] the BST bounds in [9] can be updated to 16.11 for arbitrary metric spaces, and to 12.53 for the rectilinear plane (see Table 1) In this paper we introduce a new approach to these problems, based on zero skew stretching of ....

.... strongly polynomial Runtime in this paper O(n 2 ) O(nlogn) O(n 2 ) O(nlogn) Table 1: Summary of results and comparison to results of Charikar et al. 9] Values marked with asterisks are computed by taking in account the currently best Steiner tree approximation of Robins and Zelikovsky [22] and Arora s PTAS for computing rectilinear Steiner trees [1, 2] 2 Constructive lower bounds In this section, we establish new lower bounds for the ZST and BST problems in an arbitrary metric space; in contrast to the lower bounds of Charikar et al. 9] these bounds are constructive. The ....

ROBINS, G., AND ZELIKOVSKY, A. Improved Steiner tree approximation in graphs. In Proc. 11th ACM-SIAM Symp. on Discrete Algorithms (2000), pp. 770--779. 10

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G. Robins and A. Zelikovsky. Improved Steiner Tree Approximation in Graphs. In Proc. 10th Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 770-779, 2000.

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Robins, G. and Zelikovsky, A.: Improved Steiner tree approximation in graphs. Proceedings of the 11th annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, (2000) 770--779.

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th annual ACM-SIAM symposium on Discrete algorithms, pages 770--779, 2000.

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th annual ACM-SIAM symposium on Discrete algorithms, pages 770--779, 2000.

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G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. In Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, pages 770--779,

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G. Robins and A. Zelikovsky, Improved Steiner tree approximation in graphs. SODA 2000.

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proc. Symposium on Discrete Algorithms, pages 770779, 2000.

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G. Robins and A. Zelikovsky, "Improved Steiner tree approximation in graphs," Proc of ACM-SIAM Symp Discrete Algorithms (SODA

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G. Robins and A. Zelikovsky, "Improved Steiner Tree Approximation in Graphs," in Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2000.

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Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770--779, 2000.

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Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, 2000, pp. 770-779.

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G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. Proceedings of the 10th ACM-SIAM SODA, 2000. 9

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G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 770-779, 2000.

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G. Robins, A. Zelikovsky, Improved Steiner tree approximation in graphs, In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2000, 770--779.

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G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. In 11th ACM Symposium on Parallel Architectures and Algorithms, pages 770--779, 2000.

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G. Robins and A. Zelikovsky, "Improved Steiner Tree Approximation in Graphs," in Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2000.

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G. Robins, A. Zelikovsky, Improved Steiner tree approximation in graphs, In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms 2000, pp. 770--779.

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Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms 2000, 770-779.

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G. Robins and A. Zelikovsky. Improved Steiner Tree Approximations in Graphs. In Proc. of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00), pp. 770-779, 2000.

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Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770--779, 2000.

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G. Robins and A. Zelikovsky, Improved Steiner tree approximation in graphs, Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2000, 770-779.

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G. Robins and A. Zelikovsky, \Improved Steiner tree approximation in graphs," in Proc. 11th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 770-779, 2000.

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proc. 11th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 770--779, 2000.

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Robins, G., Zelikovsky, A.: Improved Steiner Tree Approximation in Graphs, in Proc. 10th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp:770-779, 2000.

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Gabriel Robins and Alexander Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770--779, 2000.

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G. Robins and A. Zelikovsky, \Improved steiner tree approximation in graphs," in Proceedings of ACM/SIAM Symposium on Discrete Algorithms, 2000, pp. 770-779.

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G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 10th ACM-SIAM symposium on Discrete Algorithms, pages 770--779, 1999.

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