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{1,2,∞}. We present a polynomial algorithm for the problem when the graph induced by the edges with 
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(), there can be no polynomial time approximation algorithm for the problem that produces a solution with upgrading cost at most α < ln times the optimal upgrading cost even if the budget can be violated by a factor (), for any polynomial time computable function (). This result continues to hold, with () = being any polynomial, even when the difference between the maximum and minimum edge weights is bounded by a polynomial in .• Finally, we show that using a sample binary search over the set of admissible values, the dual problem can be solved with an appropriate performance guarantee.
>0, the approximation scheme finds a (1+
1/
) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space.
doi:10.1016/j.orl.2004.05.005 
Copyright © 2004 Elsevier B.V. All rights reserved.
Approximating k-hop minimum-spanning trees*1
Ernst Althaus
, a, 1, Stefan Funke, b, 2, Sariel Har-Peled, c, Jochen Könemann, d, 3, Edgar A. Ramos, c and Martin Skutella
, , b
Received 14 April 2004;
accepted 12 May 2004.
Available online 10 July 2004.
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Abstract
Given a complete graph on˜n nodes with metric edge costs, the minimum-cost k-hop spanning tree (kHMST) problem asks for a spanning tree of minimum total cost such that the longest root-leaf-path in the tree has at most k edges. We present an algorithm that computes such a tree of total expected cost O(log n) times that of a minimum-cost k-hop spanning-tree.
Author Keywords: Approximation algorithms; Minimum spanning trees; Depth restriction; Metric space approximation
