M. Karpinsky and A. Zelikovsky. New approximation algorithms for the steiner tree problem. In Technical Report, Electronic Colloquium on Computational Complexity (ECCC) TR95-030, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....paper, the underlying network links are directed, and so, we are interested in the directed Steiner tree problem. Most of the theoretical work in the existing literature has focussed on undirected versions of the problem, and algorithms achieving constant factors have been given by several authors [7, 15]. Work has also been done on the performance of these undirected Steiner tree heuristics in the context of multicast routing [22] The problem of multicast routing in directed networks and the directed Steiner tree problem has received considerable attention only recently [10, 20] 5. ....

M. Karpinsky and A. Zelikovsky. New approximation algorithms for the Steiner tree problem. Technical Report, Electronic Colloquium on Computational Complexity (ECCC), TR95-030, 1995.

....1:35 ln n. 1. Introduction The Steiner tree problem is a classical problem in networks, and is of wide interest. The problem is known to be NP hard for graphs, as well as in most metrics [5] Much effort has been devoted to the study of polynomial time approximation algorithms for this problem [1, 8, 9, 10, 13]. The Steiner tree problem is defined as follows: given a graph G = V; E) and a subset of vertices S V we wish to compute a minimum weight tree that includes all the vertices in S. The tree may include other vertices not in S as well. The vertices in S are also called terminals (sometimes ....

....of vertices S V we wish to compute a minimum weight tree that includes all the vertices in S. The tree may include other vertices not in S as well. The vertices in S are also called terminals (sometimes these are referred to as required vertices) For results on edge weighted problems see [14, 2, 1, 10]. In A preliminary version of this paper has been submitted to the 1998 Integer Programming and Combinatorial Optimization Conference. y Part of this work was done while S. Guha was at the University of Maryland and his research was supported by NSF Research Initiation Award CCR 9307462. ....

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem ", Journal of Combinatorial Optimization, 1(1):47--66, (1997).

....known as Steiner vertices) The Steiner tree problem is NP Complete even when the graph is induced by points in the plane [10] and is MAX SNP hard [3] for general graphs. The undirected version has been very well studied and algorithms achieving constant factors have been given by several authors [17, 23, 26, 2, 19]. The directed version of the Steiner tree problem is a natural extension of the undirected version and is defined as follows. Given a directed weighted graph G = V; A) a specified root r 2 V , and a set of terminals X V (jXj = k) the objective is to find the minimum cost arborescence rooted ....

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem ", Technical Report, Electronic Colloquium on Computational Complexity (ECCC): TR95030 (1995). 14

....and Berman [14] and by Takahashi and Matsuyama [19] These algorithms achieved an approximation factor of 2 Gamma 1 k where k is the number of terminals. Zelikovsky [22] obtained a ratio of 11 6 and this has been further improved by Berman and Ramaiyer [3] and Karpinski and Zelikovsky [16]. The best ratio achievable currently is 1:644. In a recent result, Arora [2] has given a polynomial time approximation scheme for the Steiner tree problem on graphs induced by points in the plane. The directed version of the Steiner tree problem is a natural extension of the undirected version ....

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem ", Technical Report, Electronic Colloquium on Computational Complexity (ECCC): TR95030 (1995).

....known as Steiner vertices) The Steiner tree problem is NP Complete even when the graph is induced by points in the plane [10] and is MAX SNP hard [3] for general graphs. The undirected version has been very well studied and algorithms achieving constant factors have been given by several authors [17, 23, 26, 2, 19]. The directed version of the Steiner tree problem is a natural extension of the undirected version and is de ned as follows. Given a directed weighted graph G = V; A) a speci ed root r 2 V , and a set of terminals X V (jXj = k) the objective is to nd the minimum cost arborescence rooted at ....

M. Karpinsky and A. Zelikovsky, \New approximation algorithms for the Steiner tree problem ", Technical Report, Electronic Colloquium on Computational Complexity (ECCC): TR95030 (1995). 14

....(1993) suggested to use a minimum spanning tree heuristic that, for n sequences, satisfies a worst case ratio of Gamma 2 Gamma 2 n Delta compared to the optimum for n sequences. Other recent approaches achieve even better worst case ratios. An example is the heuristic of Du et al. 1991) Karpinski and Zelikovsky (1997) achieve the currently best known bound of approximately 1:644. Alternative methods (Berman and Ramaiyer 1994) have also been suggested (Jiang et al. 1994) Apart from the implementation and complexity issues associated with the practical application of the algorithms of Du et al. 1991) and ....

....achieve the currently best known bound of approximately 1:644. Alternative methods (Berman and Ramaiyer 1994) have also been suggested (Jiang et al. 1994) Apart from the implementation and complexity issues associated with the practical application of the algorithms of Du et al. 1991) and Karpinski and Zelikovsky (1997) there is another issue concerning the topology of the results. Let us review some concepts that are necessary for the analysis. Usually the topology of a Steiner tree T in a graph is identified with the following tree T . Nodes of T are the given nodes and the inner nodes of T with degree ....

Karpinski, M., and Zelikovsky, A. 1997. New approximation algorithms for the Steiner tree problems. Journal of Computational Optimization 1, 1--19.

.... a given set of terminals in a graph or network can be approximated in polynomial time up to a factor of 2, cf. e.g. Choukhmane [6] or Kou, Markowsky, Berman [14] After a long period without any progress Zelikovsky [23] Berman and Ramaiyer [2] Zelikovsky [24] and Karpinski and Zelikovsky [13] improved the approximation factor step by step from 2 to 1:644. In this paper we present RNC approximation algorithm for the Steiner problem with approximation ratio (1 ffl) 5=3 for all ffl 0. The running time of these algorithms is polynomial in ffl Gamma1 and n. Our algorithms also give ....

....relative greedy heuristic for approximating mst(H r (N; K) that yields an approximation algorithm for the length of a Steiner minimum tree with performance ratio 1 ln 2 1:693. A slight further improvement, then, led to a ratio of 1:644 for any positive 0, see Karpinski and Zelikovsky [13]. In order to use the algorithm of the previous section for solving the spanning tree problem in H 3 (N; K) we have to reduce the spanning tree problem in hypergraphs with edges containing at most three vertices to a corresponding problem in a 3 uniform hypergraph, i.e. a hypergraph where all ....

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M. Karpinski and A.Z. Zelikovsky, New approximation algorithms for the Steiner tree problem, Journal of Combinatorial Optimization 1 (1997), 47--65.

....a 3 ln k approximation for finding a connected dominating set for a specified subset of vertices. We also outline a second algorithm that gives an approximation factor of (1 c)H(min( Delta; k) O(1) where c is the best approximation ratio for the Steiner tree problem (currently c = 1:644 [12]) Even though this algorithm has a better approximation guarantee, it is not practical due to the high running time, albeit polynomial. 1.2 Preliminaries The Steiner tree problem is defined as follows: given a subset of required vertices in an edge weighted graph, find a minimum weight tree ....

....it. Since members of the optimum connected dominating set al..ong with the members of the dominating set we found, form a spanning tree, we can prove a performance guarantee of c(1 H ( Delta) where c is the best approximation ratio for the unweighted Steiner tree problem (currently c = 1:644 [12]) For the special case when the required vertices form a dominating set in a graph and all edges have unit weight, Berman and Furer [3] have announced a new algorithm with c = 4 3 . Thus we can improve the performance ratio to 4 3 (1 H ( Delta) By applying a simple greedy strategy to ....

[Article contains additional citation context not shown here]

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem ", Technical Report, Electronic Colloquium on Computational Complexity (ECCC): TR95-030, (1995).

....1:35 ln n. 1. Introduction The Steiner tree problem is a classical problem in networks, and is of wide interest. The problem is known to be NP hard for graphs, as well as in most metrics [5] Much effort has been devoted to the study of polynomial time approximation algorithms for this problem [1, 8, 9, 10, 13]. The Steiner tree problem is defined as follows: given a graph G = V; E) and a subset of vertices S V we wish to compute a minimum weight tree that includes all the vertices in S. The tree may include other vertices not in S as well. The vertices in S are also called terminals (sometimes ....

....of vertices S V we wish to compute a minimum weight tree that includes all the vertices in S. The tree may include other vertices not in S as well. The vertices in S are also called terminals (sometimes these are referred to as required vertices) For results on edge weighted problems see [14, 2, 1, 10]. In this paper, we concentrate on the study of the node weighted version, where the nodes, rather than Part of this work was done while S. Guha was at the University of Maryland and his research was supported by NSF Research Initiation Award CCR 9307462. Email addr: sudipto cs.stanford.edu y ....

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem", Journal of Combinatorial Optimization, 1(1):47--66, (1997).

.... are similar to the ones for MIN TSP: exact optimization is NP hard in R 2 both in the Rectilinear ( 1 ) case [GJ77] and in the Euclidean ( 2 ) case [GGJ77] Constant factor approximation is achievable in any metric space (the best factor should be 1:644 due to Karpinski and Zelikovsky [KZ95] in general metric spaces the problem is Max SNP hard [BP89] Arora s algorithm achieves approximation (1 ffl) in R d in time n O( log d Gamma2 n) ffl d Gamma1 ) No non approximability result was known for geometric versions of the problem. Our Results. We prove the Max ....

M. Karpinski and A. Zelikovsky. New approximation algorithms for the Steiner tree problems. Technical Report ECCC TR95-030, 1995.

....known as Steiner vertices) The Steiner tree problem is NP Complete even when the graph is induced by points in the plane [9] and is MAXSNP hard [4] for general graphs. The undirected version has been very well studied and algorithms achieving constant factors have been given by several authors [15, 21, 24, 3, 17]. The directed version of the Steiner tree problem is a natural extension of the undirected version and is defined as follows. Given a directed graph G = V; A) a specified root r 2 V , and a set of terminals X V (jXj = k) the objective is to find the minimum cost arborescence rooted at r and ....

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem", Technical Report, Electronic Colloquium on Computational Complexity (ECCC): TR95-030 (1995).

....a connected dominating set. For the unweighted case, we provide an approximation algorithm that runs in polynomial time in Subsection 4.2. It has an approximation factor of (1 c)H(min( Delta; k) c Gamma 1, where c is the approximation ratio for the Steiner tree problem (currently c = 1:644 [15]) and k is the size of the set we want to dominate. When the vertices have weights, the Steiner connected dominating set problem is at least as hard as the notorious set TSP problem on graphs (defined shortly) for which no non trivial approximation algorithms are known. The same is true for the ....

....it. Since members of the optimum connected dominating set al..ong with the members of the dominating set we found, induce a connected subgraph, we can prove an approximation ratio of c(1 H ( Delta) where c is the approximation ratio for the unweighted Steiner tree problem (currently c = 1:644 [15]) For the special case when the required vertices form a dominating set in a graph and all edges have unit weight, Berman and Furer [3] have announced a new algorithm with c = 4 3 . Thus we can improve the approximation ratio to 4 3 (1 H ( Delta) by using their algorithm. By applying a ....

[Article contains additional citation context not shown here]

M. Karpinsky and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem ", Technical Report, Electronic Colloquium on Computational Complexity (ECCC): TR95-030, (1995).

....deal. We know by now, that unless P=NP, some problems, such as CLIQUE cannot be approximated in polynomial time even within a factor n 1 Gammaffl for any ffl 0 (cf. Hastad [H96] Some other problems like MAX CUT (cf. Goemans and Williamson [GW94] or STEINER TREE (cf. Karpinski and Zelikovsky [KZ97a]) can be approximated to within some small fixed constant factor. Till recently only a very few optimization problems were known to have polynomial time approximation schemes (PTAS) approximating to within arbitrary small constant factors. Some of the approximation algorithms with small ....

....problems were known to have polynomial time approximation schemes (PTAS) approximating to within arbitrary small constant factors. Some of the approximation algorithms with small approximation ratios achieve also good practical performances, like some cases of STEINER TREE problems (cf. [KZ97a]) some other algorithms do not yield yet efficient practical methods of dealing with optimization problems. In this paper we are concerned with the problem of efficient approximability of the dense instances of NP hard optimization problems. Recently, the first polynomial time approximation ....

[Article contains additional citation context not shown here]

M. Karpinski and A. Zelikovsky, New Approximation Algorithms for the Steiner Tree Problem, J. of Combinatorial Optimization 1 (1997), pp. 47-65.

....a tree from the subgraph obtained. This heuristic can be implemented in time O(jV j 2 ) M88] The minimum spanning tree heuristic gives a 2 approximation for the Steiner tree problem [TM80] and the best up today polynomial time approximation guarantee of Karpinski Zelikovsky is about 1. 644 [KZ97]. Unfortunately, the minimum APPROXIMATING DENSE CASES OF COVERING PROBLEMS 5 spanning tree heuristic has the same performance ratio of 2 for the dense Steiner tree problem. Note that for ffl 1 2 , ffl dense Steiner tree problem is a sub case of the network Steiner tree problem with ....

M. Karpinski and A. Zelikovsky, New approximation algorithms for the Steiner tree problem. Journal of Combinatorial Optimization 1: 47--65, 1997.

....points. Using MSTH we can find an optimal Steiner tree if we add all Steiner points to the terminal set. Remark 1 An optimal Steiner tree can be found exactly in O(n k ) time. MSTH gives 2 approximation for STP [9] and the best up today polynomial time approximation guarantee is about 1. 644 [7]. From the other side, STP is known to be MAX SNP complete [4] In the B sparse STP the degree of any vertex is bounded by a constant B. It is known that STP in the rectilinear metric (a sub case of 4 sparse STP) is NP complete but the question whether it is MAX SNP hard or not is still open. In ....

M. Karpinski and A. Zelikovsky. New approximation algorithms for the Steiner tree problem. Journal of Combinatorial Optimization 1: 1--19, 1997.

No context found.

M. Karpinsky and A. Zelikovsky. New approximation algorithms for the steiner tree problem. In Technical Report, Electronic Colloquium on Computational Complexity (ECCC) TR95-030, 1995.

No context found.

M. Karpinsky and A. Zelikovsky. New approximation algorithms for the steiner tree problem. In Technical Report, Electronic Colloquium on Computational Complexity (ECCC) TR95-030, 1995.

No context found.

M. Karpinski and A. Zelikovsky, "New Approximation Algorithms for the Steiner Trees Problems", Journal of Combinatorial Optimization, 1 (1997), 47-65.

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