Abstract
We consider the approximation characteristics of constrained spanning and Steiner tree problems in weighted undirected graphs where the edge costs and delays obey the triangle inequality. The constraint here is in the number of hops a message takes to reach other nodes in the network from a given source. A hop, for instance, can be a message transfer from one end of a link to the other. A weighted hop refers to the amount of delay experienced by a message packet in traversing the link. The main result of this chapter shows that no approximation algorithm for a delay-constrained spanning tree satisfying the triangle inequality can guarantee a worst case approximation ratio better than Θ(log n) unless NP ⊂ DTIME(n log logn). This result extends to the corresponding problem for Steiner trees which satisfy the triangle inequality as well.
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Acknowledgements
The author benefited from discussions with Matt Stallmann of North Carolina State University. Support from the Sir Ross and Sir Keith Smith Foundation is gratefully acknowledged. The comments from the referee were particularly helpful. Since the early 1990s, the online compendium of Crescenzi and Kann, and more recently, their book [1], has been a great help to the research community.
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Manyem, P. (2009). Constrained spanning, Steiner trees and the triangle inequality. In: Pearce, C., Hunt, E. (eds) Optimization. Springer Optimization and Its Applications, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-0-387-98096-6_19
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DOI: https://doi.org/10.1007/978-0-387-98096-6_19
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