# A branch-and-bound algorithm for computing node weighted steiner minimum trees

## Abstract

Given *n* regular points in the Euclidean plane, the node-weighted Steiner minimum tree (NWSMT) is a straight line network interconnecting these *n* regular points and some Steiner points with a minimum cost, where the cost of the network is the sum of the edge lengths plus the total cost of the Steiner points. In 1995, [11] proved that a tight upper bound on the maximum degree of *Steiner points* in a NWSMT is 4. In 1996, [14] used this result to propose a modified Melzak procedure for computing a NWSMT. However, that procedure requires exponential time to compute a minimum cost network under a given topology. In this paper, we prove that there exists a NWSMT in which the maximum degree of *regular points* is no more than 5 and that this upper bound is tight. For a given topology interconnecting *n* regular points, we show that the Xue-Ye algorithm [15] for minimizing a sum of Euclidean norms can be used to compute an (1 + *ε*)-approximation of the minimum cost network in *n*^{1.5}(log *n* + log 1/*ε*) time for any positive *ε*. These results enable an algorithm that computes a NWSMT by enumerating all the possible Steiner topologies. We prove a bounding theorem that can be used in a branch-and-bound algorithm and present preliminary computational experience.

## Keywords

Node weighted Steiner minimum trees maximum node degrees minimum cost network under a given topology branch-and-bound## Preview

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