A lower bound for the steiner tree problem in directed graphs

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Abstract
In this article, we introduce a new integer programming formulation for the minimum Steiner tree problem in directed graphs. With the observation that every Steiner tree contains a two‐terminal Steiner tree for every pair of the terminals, our formulation is based on the linear programming formulation for the two terminal Steiner tree polyhedron obtained by Ball et al. [2]. By the results of Ball et al. [2], this formulation contains a large class of facets that are different from those induced by the well‐known Steiner cut constraints. We give a general form of the dual ascent algorithm and discuss the relationship between this algorithm and the projection method for extended formulations. This dual ascent algorithm is applied to the new formulation to obtain a lower bound for the minimum Steiner tree problem. In the algorithm, we use the dual ascent algorithm introduced by Wong [16] as a subroutine and improve his lower bound. Some computational results are given in Section 3.

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