A Constant-Factor Approximation for Quasi-bipartite Directed Steiner Tree on Minor-Free Graphs
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We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for K_r-minor-free graphs our approximation guarantee is O(r·√(log r)) and, further, for planar graphs our approximation guarantee is 20. Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the Directed Steiner Tree instances we consider.
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