M. Karpinski and A. Zelikovsky. New approximation algorithms for the Steiner tree problems. J. Comb. Opti., 1:4765, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....this problem is known to be in P[15] However, if the weights are given in unary, the problem is in RP[5] and even in RNC[16] This gives the Promel Steger result. For other algorithms for this problem see [3, 13] An algorithm achieving a slightly better approximation factor of 1. 644 appears in [14]. However, it is too involved in its current form for this survey; moreover, to beat the factor of 5=3, it takes time exceeding O(n 20 ) 3 Steiner networks via LP rounding The Steiner network problem generalizes the metric Steiner tree problem to higher connectivity requirements: Given a ....

M. Karpinski and A. Zelikovsky. New approximation algorithms for the Steiner tree problem. Electr. Colloq. Comput. Compl., TR95-030, 1995. 11

.... 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.

....for . For large , we note that the problem of computing the dispersion of an arbitrary set of points in is equivalent to the Steiner tree problem in a grid, under the metric. The latter problem is known to be NP complete [10] A polynomial time approximation algorithm for this problem is given in [18], with the approximation ratio of . The best known exact algorithms may be found in [23] 24] These algorithms make it possible to compute the dispersion of up to 2000 points. B. Lattice Interleavers Lattice interleavers, which are interleaving schemes based on lattices, will play an important ....

M. Karpinski and A. Zelikovsky, "New approximation algorithms for the Steiner tree problem," J. Comb. Optimiz., vol. 1, pp. 1--19, 1997.

....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] ....

M. Karpinski, A. Zelikovsky, New approximation algorithms for the Steiner tree problems, J. Comb. Optim. 1 (1997), 47--65.

....to vary metric for a better bound Could we forget the greedy idea and design a better approximation with only variable metric idea Answering these questions requires deeper understanding the variable metric method. We attempt to obtain new algorithms from this study. Karpinski and Zelikovsky [62] proposed a preprocessing procedure to improve existing better approximations. First, they use this procedure to choose some Steiner points and then run a better approximation algorithm on the union of the set of regular points and the set of chosen Steiner points. This preprocessing improves the ....

....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 ....

M. Karpinski and A.Z. Zelikovsky, New approximation algorithms for Steiner tree problems, Journal of Combinatorial Optimization 1 (1997) 4765.

....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] ....

M. Karpinski, A. Zelikovsky, New approximation algorithms for the Steiner tree problems, J. Comb. Optim. 1 (1997), 47--65.

....this problem 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 is 1. 644 due to Karpinski and Zelikovsky [KZ97] in general metric spaces the problem is Max SNP hard [BP89] Arora [Aro96, Aro98] Mitchell [Mit97] and Rao and Smith [RS98] show how to extend their geometric TSP approximation schemes to geometric Min ST. The running time of these approximation schemes are the same as reported in the ....

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

....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 ....

Marek Karpinski and Alexander Zelikovsky. New approximation algorithms for the Steiner tree problems. Journal of Combinatorial Optimization, 1:47--65, 1997.

....By applying this framework to one specific algorithm we obtain a new polynomial time approximation algorithm for the Steiner tree problem in graphs that achieves a performance ratio of 1.598 after 11 iterations. This beats the so far best known factor of 1. 644 due to Karpinski and Zelikovsky [10]. With the help of a computer program we estimate the limit performance of our algorithm to be 1.588. 1 Introduction Given a graph G = V; E) a subset T ae V of terminals and a length function c : E R on the edges of G, then the Steiner Tree Problem asks for a shortest network connecting the ....

....cannot exist a polynomial time approximation scheme for this problem. Here we will present a polynomial time approximation algorithm for the Steiner tree problem in graphs which achieves a performance ratio of 1.598. This beats the so far best known factor of 1. 644 due to Karpinski and Zelikovsky [10]. The new idea of our algorithm is to iteratively use a parameterized Steiner tree algorithm to improve the solution found so far by the algorithm. This is done by successively adding in a certain way additional terminals to the set of terminals given in the beginning. While this clearly worsens ....

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

.... 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 ....

.... replace a full component K with C[K] then (i) it will not cost anything to add a full component K in the overall solution and (ii) we decrease the gain of full components which disagree with K by a small value (e.g. less than in the Berman Ramayer algorithm for large k, and much smaller than in [15] for any k) Our algorithm iteratively modifies a terminal spanning tree T , which is originally MST (G S ) by incorporating into T loss contracted full components greedily chosen from G. The intuition behind the gain over loss objective ratio is as follows. The cost of the approximate solution ....

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

....a great 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 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 A preliminary version of this paper appeared in Proc. Randomized and Approximation Techniques in Computer Science, LNCS 1296, Springer, 1997, pp 1 14. Dept. of Computer Science, University of Bonn, 53117 Bonn, Email: marek cs.unibonn. ....

....constant factors. In a recent breaktrough Arora [A96] shown existence of PTASs for Euclidean TSP and a number of other geometric problems. 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 solutions for 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. Starting in 1995, the first polynomial time ....

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

.... 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. ....

.... replace a full component K with C[K] then (i) it will not cost anything to add a full component K in the overall solution and (ii) we decrease the gain of full components which disagree with K by a small value (e.g. less than in Berman Ramayer s algorithm for large k and much smaller than in [14] for any k) Our algorithm iteratively modifies a terminal spanning tree T , which is originally MST (G S ) by incorporating into T loss contracted full components greedily chosen from G. The intuition behind the gain over loss objective ratio is as follows. The cost of the approximate solution ....

[Article contains additional citation context not shown here]

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

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M. Karpinski and A. Zelikovsky. New approximation algorithms for the Steiner tree problems. J. Comb. Opti., 1:4765, 1997.

No context found.

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

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