On the approximability of the Steiner tree problem in phylogeny

AbstractThree results on the Steiner tree problem are presented: (i) Computing optimum k-restricted Steiner tree is APX-complete for k ⩾4, (ii) the minimum-cost k-restricted Steiner tree problem in phylogeny is APX-complete for k ⩾4, and (iii) the k-Steiner ratio for the Steiner tree problem in phylogeny matches the corresponding ratio for metric spaces defined on networks. The first two results suggest that there is a limit to the approximability of the optimum solution to the k-restricted Steiner tree problem. The results are significant because k-restricted trees are used in approximation algorithms for the minimum Steiner tree problem. The k-Steiner ratio, which establishes a relation between the size of the optimum Steiner tree and the size of the optimum k-restricted tree, arises in the analysis of many of the same algorithms