A Faster Approximate Minimum Steiner Tree Algorithm & Primal-Dual and Local-Ratio Greedy Algorithms

We prove matching upper and lower bounds for the Minimum Steiner Tree (MStT) problem within the adaptive priority stack model introduced by Borodin, Cashman, and Magen [2] and lower bounds within the adaptive priority queue model introduced here. These models, which are extensions of the adaptive priority model of Borodin, Nielson, and Rackoff [3], are said to capture reasonable primal-dual and local-ratio extensions of greedy algorithms. AB: I am worried about how to state the rest of this abstract since the tight bound depends on the input model. I recommend that we just omit the rest of the abstract and wait to the intro to state results. Borodin etal. [2] proved a 4 3 lower bound on the approximation ratio obtainable for the problem in the stack model. We improve this to 2−O( 1 |V | ). This and similar results prove that the above upper bound is tight, that the adaptive priority stack model is incomparable with the fully adaptive priority branching tree (pBT) model, and that the online stack model is incomparable with the adaptive priority model. We also prove a lower bound in the priority model for complete graphs improving the result of Davis and Impagliazzo [5] from 1 1 4 to 1 1 3 . The priority queue model is better because ????? Within it we prove a 7 6 approximation lower bound.