A. Zelikovsky, Better approximation bounds for the network and Euclidean Steiner tree problems, technical report CS-96-06, University of Virginia, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Germany, groepl, hougardy, nierhoff, proemel informatik.hu berlin.de. It has been known 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 ....

....so that it might indicate an alternative approach to the general case. As a by product, this method allows for a simple instance that shows that the performance ratio of 73=60 is tight. Such instances are not yet known for the other approximation algorithms for the Steiner tree problem in graphs [18, 3, 19, 11, 9]. Lower bounds for the performance ratio of some of these algorithms are given in [7] A slightly more general case are 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] ....

A. Zelikovsky, Better approximation bounds for the network and Euclidean Steiner tree problems, technical report CS-96-06, University of Virginia, 1996.

....Du and Hwang [23] proved this conjecture and thus showed that the MST is a 2 # 3 approximation to the optimum Steiner tree. A spate of research activity in recent years starting with the work of Zelikovsky[65] has provided better approximation algorithms, with an approximation ratio around 1. 143 [66]. The metric case does not have an approximation scheme if P # NP [16] The Steiner Tree problem involves an objective function that is a sum of edge lengths and it obeys the Patching Lemma (as is easily checked) Now we briefly describe the algorithm. First we perturb the instance to ensure ....

A. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Tech. Rep. CS-96-06, University of Virginia, 1996. 30

....heuristic exists in any metric space satisfying the following conditions: 1) The Steiner ratio is smaller than one. 2) The Steiner minimum tree on any fixed number of points can be computed in polynomial time. These metric spaces include Euclidean plane and Euclidean spaces. Zelikovsky [104] used a different potential function in his greedy approximation and obtained an approximation with performance ratio satisfying PR ae Gamma1 k (1 Gamma ln ae 2 ) Although Zelikovsky s idea starts from a point different from Chang s one, the two approximations are actually similar. To see ....

....also give approximations for Steiner minimum trees with performance ratio approach to the inverse of the 3 Steiner ratio. This probably is the best possible performance ratio. Thus, the conjecture of Arora et al. is an attractive problem to our further research. A more accurate analysis [104, 62, 102] for the performance ratios of BermanRamaiyer s algorithm and Karpinski Zelikovsky s preprocessing requires bounds for t k and a similar number t k . The techniques in [15, 16] for determining the k Steiner ratio seems very promising for establishing tight upper bounds for t k and t k . The ....

A.Z. Zelikovsky, Better approximation bounds for the network and Euclidean Steiner tree problem, manuscript, 1995. 27

....Pr 296 6 1. Address of all authors: Institut fr Informatik, Humboldt Universit t zu Berlin, 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 ....

....so that it might indicate an alternative approach to the general case. As a by product, this method allows for a simple instance that shows that the performance ratio of 73 60 is tight. Such instances are not yet known for the other approximation algorithms for the Steiner tree problem in graphs [3, 9, 11, 18, 19]. Lower bounds for the performance ratio of some of these algorithms are given in [7] A slightly more general case are quasi bipartite 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] ....

A. Zelikovsky, Better approximation bounds for the network and Euclidean Steiner tree problems, technical report CS-96-06, University of Virginia, 1996. 6

....Gilbert Pollak [26] conjecture said it can t be any further from the optimum; this conjecture was recently proved by Du and Hwang [19] A spate of research activity in recent years (starting with the work of Zelikovsky[64] has provided better algorithms, with an approximation ratio around 1. 143 [65]. The metric case is MAX SNP hard [13] k TSP: Given n nodes in # d and an integer k 1, find the shortest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [46] K # 8) is PLS complete [41] This strongly suggests that no polynomial time algorithm can ....

A. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Tech. Rep. CS-96-06, University of Virginia, 1996.

.... and Thurimella, 1993) Steiner Tree: Given an undirected graph with positive edge weights and a subset of the vertices called terminals, find a minimum weight set of edges through which all the terminals (and possibly other vertices) are connected (Zelikovsky, 1993; Berman and Ramaiyer, 1994; Zelikovsky, 1996; Hochbaum, 1995, Ch. 8) Steiner Forest: Given a weighted graph and a collection of groups of terminals, find a minimumweight set of edges through which every pair of terminals within each group are connected (Agrawal et al. 1995; Hochbaum, 1995, Ch. 4) The algorithm for this problem is based ....

Zelikovsky, A. (1996). Better approximation bounds for the network and Euclidean Steiner tree problems. Technical Report CS-96-06, Department of Computer Science, University of Virginia. Mon, 25 Mar 1996 16:47:16 GMT.

....trees [17] to the Generalized Tree Alignment Problem. Gusfield [11] suggested to use a minimum spanning tree heuristic that, for n sequences, satisfies a worst case ratio of (2 Gamma 2 n ) compared to the optimum. But this strategy like other heuristics, see Du et al. 6] and Zelikovsky [27] share the same shortcoming with respect to our application: The final trees are assembled by linking previously computed subtrees with k or less nodes via given sequences. As a consequence, for larger input sets, given sequences occur as inner nodes in the resulting tree. This fact makes it ....

A. Zelikovsky. Better Approximation Bounds for the Network and Euclidean Steiner tree problems. Technical report 96-06, Department of Computer Science, U. of Virginia, 1996.

....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 ....

Alexander Zelikovsky. Better approximation bounds for the network and euclidean Steiner tree problems. Technical Report CS-96-06, University of Virginia, 1996. 8

....Gilbert Pollak [25] conjecture said it can t be any further from the optimum; this conjecture was recently proved by Du and Hwang [18] A spate of research activity in recent years (starting with the work of Zelikovsky[63] has provided better algorithms, with an approximation ratio around 1. 143 [64]. The metric case is MAX SNP hard [13] k TSP: Given n nodes in # d and an integer k 1, find the shortest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [45] achieves a constant factor approximation in # 2 . k MST: Given n nodes in # d and an ....

A. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Tech. Rep. CS-96-06, University of Virginia, 1996.

....result of Borchers and Du [5] to [4] The value given in [4] is 1.746) All performance ratios given in this paper are rounded up to the third digit. Year Performance Authors Ratio 1980 2.000 Takahashi, Matsuyama [12] 1993 1.834 Zelikovsky [13] 1994 1.734 Berman, Ramaiyer [4] 1995 1. 694 Zelikovsky [14] 1997 1.667 Promel, Steger [11] 1997 1.644 Karpinski, Zelikovsky [10] 1998 1.598 Hougardy, Promel [this paper] Table 1: Steiner tree approximation algorithms Takahashi and Matsuyama [12] were the first proving that the well known minimum spanning tree heuristic achieves a performance ratio of 2. ....

....for some small values of k for optimally chosen values of ff i . k ratio 1 1.694 2 1.644 3 1.626 4 1.616 5 1.611 6 1.607 7 1.604 . 11 1.598 12 1.597 . 1 1. 588 Table 2: Performance ratio of k IRGH Note that in the special case k = 1 we obtain the algorithm of Zelikovsky [14] while for k = 2 we obtain the algorithm of Karpinski and Zelikovsky [10] In the latter case ff 1 has to be chosen as 0:436. For k = 3 one has to choose ff 1 = 0:698 and ff 2 = 0:248. In the case k = 11 the sequence (ff 1 ; ff k ) looks as follows: 1:365; 1:026; 0:792; 0:615; 0:474; ....

A. Zelikovsky, Better approximation bounds for the network and Euclidean Steiner tree problems, Technical report CS-96-06, University of Virginia.

....[BFK 94, BR94] Most recently, Karpinski and Zelikovsky [KZ94] added a preliminary phase to Berman Ramaiyer, which without increasing the running time further reduces the approximation ratio to 19 15 # 1.267 for k = 4. 8.3.2. 4 The relative greedy algorithm Very recently, Zelikovsky [Zel94] modified his original greedy algorithm by changing the way full components are chosen. The result is a sequence of algorithms with approximation ratios converging to 1 ln# 2 . This relative greedy algorithm thus gives 1 ln2 # 1.693 for the network Steiner problem, asymptotically beating ....

A.Z. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Manuscript, 1994.

....trees [17] to the Generalized Tree Alignment Problem. Gusfield [11] suggested to use a minimum spanning tree heuristic that, for n sequences, satisfies a worst case ratio of (2 Gamma 2 n ) compared to the optimum. But this strategy like other heuristics, see Du et al. 6] and Zelikovsky [27] share the same shortcoming with respect to our application: The final trees are assembled by linking previously computed subtrees with k or less nodes via given sequences. As a consequence, for larger input sets, given sequences occur as inner nodes in the resulting tree. This fact makes it ....

A. Zelikovsky. Better Approximation Bounds for the Network and Euclidean Steiner tree problems. Technical report 96-06, Department of Computer Science, U. of Virginia, 1996.

....results, starting with important work by Zelikovsky [392] improved approximation algorithms were obtained, for both graph versions and geometric versions of the problem. In the Euclidean plane, the approximation factor has been improved to just over 1. 1 by Zelikovsky s relative greedy algorithm [393]. We refer the reader Bern and Eppstein [64] and Du and Hwang [143] for excellent surveys on these problems and the recent results. Finally, though, a PTAS was discovered by Arora [35] and Mitchell [289] This result serves to separate the geometric versions of the problem from the metric ....

A. Z. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Technical Report CS-96-06, University of Virginia, Charlottesville, VA, 1996.

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A. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Technical Report CS-96-06, Department of Computer Science, University of Virginia, 1996. 20

.... NP , the Steiner Tree Problem in general graphs cannot be approximated within a factor of 1 ffl for sufficiently small ffl 0 [4, 7] For arbitrary weighted graphs, the best Steiner approximation ratio achievable within polynomial time was gradually decreased from 2 to 1:59 in a series of works [20, 21, 2, 22, 18, 15, 10]. In this paper we present a polynomial time approximation scheme with a performance ratio approaching 1 2 1:55 which improves upon the previously best known ratio of 1.59 due to This work was supported by a Packard Foundation Fellowship, by National Science Foundation Young Investigator ....

A. Zelikovsky, "Better Approximation Bounds for the Network and Euclidean Steiner Tree Problems", Technical report CS-96-06, University of Virginia, 1996. 10

.... NP , the Steiner Tree Problem in general graphs cannot be approximated within a factor of 1 ffl for sufficiently small ffl 0 [4, 7] For arbitrary weighted graphs, the best Steiner approximation ratio achievable within polynomial time was gradually decreased from 2 to 1:59 in a series of works [19, 20, 2, 21, 17, 14, 9]. In this paper we present a polynomial time approximation scheme with a performance ratio approaching 1 ln 3 2 1:55 which improves upon the previously best known ratio of 1.59 due to Hougardy and Promel [9] We apply our heuristic to the Steiner Tree Problem in quasi bipartite graphs (i.e. ....

A. Zelikovsky, "Better Approximation Bounds for the Network and Euclidean Steiner Tree Problems", Technical report CS-96-06, University of Virginia, 1996. 10

....has at most three vertices. It was proved that performance ratios of these heuristics are at least 7 18 and 2 5 , respectively. Here we suggest several approximation algorithms for the weighted MPSP. These algorithms are applications of heuristics first suggested for the Steiner tree problem [12, 1, 2, 13]. The greedy algorithm approximates the best weighted 3 block tree in the given graph G (Section 2) In Section 3 we estimate the quality of the greedy algorithm. We prove that it Computer Science Department, University of Virginia. E mail: alexz fir.cs.virginia.edu 1 Greedy Algorithm M ....

A. Z. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems Tech. Rep. CS-96-06. University of Virginia, Charlottesville, 1996. 8

....[19, 21] is at most 11 6 1:84 and PR of Berman Ramaiyer s heuristic (BR) 2] is at most 16 9 1:78. Their run times are O(v 3 ) and O(ff v 2 n 3 ) respectively (here ff means time complexity of finding of all pairs shortest paths) The relative greedy heuristic (RGH) Zelikovsky [22]) with PR converging to 1 ln 2 1:693 asymptotically beats BR which PR converges to about 1.734 (Brochers and Du [6] In the recent paper Berman et al. 3] gave a more precise (than in the first papers [20, 2, 8] analysis of the performance ratio and runtime of BR for RSP. They proved that ....

....and our approach. In Sections 3 and 4 we describe our preprocessing of RGH and BR. Problem Heuristic Performance Ratio New PR Run time Reference NSP MST 2 O(v 2 ) 18, 16] GA 11 6 1:84 O(v 3 ) 19, 21] BR 16 9 1:78 253 144 1.757 O(v 5 ) 2] RGH 1 ln 2 ffl 1. 644 ffl polynomial [22] 1:693 ffl RSP MST 1.5 O(n log n) 11, 12] BR 61 48 1:271 19 15 1.267 O(n 1:5 ) 3] PBR 61 48 ffl 1:271 ffl 1.267 ffl O(n log 2 n) 3] 2 Gain and Loss of k Restricted Steiner Trees 2.1 Background A Steiner tree T of a set of terminals S is full if every internal node of T ....

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A. Z. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems. Manuscript, 1993.

....[18, 20] is at most 11 6 1:84 and PR of BermanRamaiyer s heuristic (BR) 2] is at most 16 9 1:78. Their runtimes are O(v 3 ) and O(ff v 2 n 3 ) respectively (here ff means time complexity of finding of all pairs shortest paths) The relative greedy heuristic (RGH) Zelikovsky [21]) with PR converging to 1 ln 2 1:693 asymptotically beats BR which converges to about 1.734 (Brochers and Du [5] In the recent paper Berman et al. [3] gave a more precise (than in the first papers [19, 2] analysis of the performance ratio of BR for RSP. They proved that its performance ratio ....

....after preprocessing. By ffl we mean existance of an algorithm for any ffl 0. Problem Heuristic Performance Ratio New PR Run time Reference NSP MST 2 O(v 2 ) 17, 15] GA 11 6 1:84 O(v 3 ) 18, 20] BR 16 9 1:78 253 144 1.757 O(v 5 ) 2] RGH 1 ln 2 ffl 1. 644 ffl polynomial [21] 1:693 ffl RSP MST 1.5 O(n log n) 10, 11] BR 61 48 1:271 19 15 1.267 O(n 1:5 ) 3] PBR 61 48 ffl 1:271 ffl 1.267 ffl O(n log 2 n) 3] In the next section we provide a synopsis of k restricted Steiner trees and our approach. In Sections 3 and 4 we describe our preprocessing of ....

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A. Z. Zelikovsky. Better approximation bounds for the network and Euclidean Steiner tree problems 1995.

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A. Zelikovsky, Better approximation bounds for the network and Euclidean Steiner tree problems, technical report CS-96-06, University of Virginia, 1996.

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