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We give the first constant-factor approximation algorithm for quasi-bipartiteinstances of Directed Steiner Tree on graphs that exclude fixed minors. Inparticular, for $K_r$-minor-free graphs our approximation guarantee is$O(r\cdot\sqrt{\log r})$ and, further, for planar graphs our approximationguarantee is 20.  Our algorithm uses the primal-dual scheme. We employ a more involved methodof determining when to buy an edge while raising dual variables since, as weshow, the natural primal-dual scheme fails to raise enough dual value to payfor the purchased solution. As a consequence, we also demonstrate integralitygap upper bounds on the standard cut-based linear programming relaxation forthe Directed Steiner Tree instances we consider.

A Constant-Factor Approximation for Quasi-bipartite Directed Steiner Tree on Minor-Free Graphs