Better approximation bounds for the network and Euclidean Steiner tree problems

The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G=(V,E,d) and Euclidean plane, respectively. For these problems, we present a series of heuristics finding approximate Steiner trees with performance guarantees coming arbitrary close to 1+ln 2= 1.693... and 1+ln(2/sqrt3) = 1.1438..., respectively. The best previously known corresponding values are close to 1.746 and 1.1546.

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