Advances in Steiner Trees pp 117-135 | Cite as

# On Approximation of the Power-*p* and Bottleneck Steiner Trees

## Abstract

Many VLSI routing applications, as well as the facility location problem involve computation of Steiner trees with non-linear cost measures. We consider two most frequent versions of this problem. In the power-p Steiner problem the cost is defined as the sum of the edge lengths where each length is raised to the power *p >* 1. In the bottleneck Steiner problem the objective cost is the maximum of the edge lengths. We show that the power-p Steiner problem is MAX SNP-hard and that one cannot guarantee to find a bottleneck Steiner tree within a factor less than 2, unless P = NP. We prove that in any metric space the minimum spanning tree is at most a constant times worse than the optimal power-p Steiner tree. In particular, for *p =* 2, we show that the minimum spanning tree is at most 23.3 times worse than the optimum and we construct an instance for which it is 17.2 times worse. We also present a better approximation algorithm for the bottleneck Steiner problem with performance guarantee log_{2} n, where n is the number of terminals (the minimum spanning tree can be 2 log_{2} n times worse than the optimum).

## Keywords

Minimum Span Tree Steiner Tree Performance Ratio Steiner Point Steiner Tree Problem## Preview

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## References

- [1]C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky,
*ProvablyGood Routing Tree Construction with Multi-Port Terminals*, in Proc. International Symposium on Physical Design, Napa Valley, CA, April 1997, pp. 96–102.Google Scholar - [2]K. D. Boese, A. B. Kahng, B. McCoy, and G. Robins,
*Near-optimal critical sink routing tree constructions*, IEEE Trans. on Comput.-Aided Des. of Integr. Circuits and Syst., 14 (1995), pp. 1417–11436.CrossRefGoogle Scholar - [3]C. Chiang, M. Sarrafzadeh, and C. K. Wong,
*Global Routing Based on Steiner Min-Max Trees*, IEEE Trans. Computer-Aided Design, 9 (1990), pp. 1318–25.CrossRefGoogle Scholar - [4]C. W. Duin and A. Volgenant,
*The partial sum criterion for Steiner trees in graphs and shortest paths*, Europ. J. of Operat. Res., 97 (1997), pp. 172–182.zbMATHCrossRefGoogle Scholar - [5]J. Elzinga, D. Hearn, and W. D. Randolph,
*Minimax multifacility location with Euclidean distances*, Transportation Science, 10 (1976), pp. 321–336.MathSciNetCrossRefGoogle Scholar - [6]J. L. Ganley,
*Geometric interconnection and placement algorithms*, PhD thesis, Dept of CS, University of Virginia, 1995.Google Scholar - [7]J. L. Ganley and J. S. Salowe,
*Optimal and approximate bottleneck Steiner trees*, Oper. Res. Lett., 19 (1996), pp. 217–224.MathSciNetzbMATHGoogle Scholar - [8]J. L. Ganley and J. S. Salowe,
*The power-P Steiner tree problem*, Nordic Journal of Computing, 5 (1998), pp. 115–127.MathSciNetzbMATHGoogle Scholar - [9]G. Hardy, E. Littlewood, and G. Polya,
*Inequalities*, Cambridge University Press, 1934.Google Scholar - [10]N. D. Holmes, N. A. Sherwani, and M. Sarrafzadeh,
*Utilization of vacant terminals for improved over-the-cell channel routing*, IEEE Trans. Computer-Aided Design, 12 (1993), pp. 780–782.CrossRefGoogle Scholar - [11]J. N. Hooker,
*Solving nonlinear multiple-facility network, location problem*, Networks, 19 (1989), pp. 117–133.MathSciNetzbMATHCrossRefGoogle Scholar - [12]R. M. Karp,
*Reducibility among combinatorial problems*, in R.E. Miller and J.W. Thatcher. Complexity of Computer Computations, 1972, pp. 85–103.CrossRefGoogle Scholar - [13]L. Lovasz and M. Plummer,
*Matching Theory*, Elsvier Science, 1986.zbMATHGoogle Scholar - [14]C. H. Papadimitriou and M. Yannakakis,
*Optimization, approximation, and complexity classes*, in Proc. ACM Symp. the Theory of Computing, 1988, pp. 229–234.Google Scholar - [15]M. Sarrafzadeh and C. K. Wong,
*Bottleneck Steiner trees in the plane*, IEEE Trans. on Computers, 41 (1992), pp. 370–374.MathSciNetCrossRefGoogle Scholar - [16]J. Soukup,
*On minimum cost networks with nonlinear costs*, SIAM J. Applied Math., 29 (1975), pp. 571–581.MathSciNetzbMATHCrossRefGoogle Scholar