On Approximation of the Power-p and Bottleneck Steiner Trees
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 log2 n, where n is the number of terminals (the minimum spanning tree can be 2 log2 n times worse than the optimum).
KeywordsMinimum Span Tree Steiner Tree Performance Ratio Steiner Point Steiner Tree Problem
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- 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
- J. L. Ganley, Geometric interconnection and placement algorithms, PhD thesis, Dept of CS, University of Virginia, 1995.Google Scholar
- G. Hardy, E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, 1934.Google Scholar
- 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