Ramin Mousavi

We present an $O(\log k)$-approximation for both the edge-weighted and node-weighted versions of \DST in planar graphs where $k$ is the number of terminals. We extend our approach to \MDST (in general graphs \MDST and \DST are easily seen to be equivalent but in planar graphs this is not the case necessarily) in which we get an $O(R+\log k)$-approximation for planar graphs for where $R$ is the number of roots.

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 $O(r\cdot\sqrt{\log r})$ 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.

For some families of distance-regular graphs such as strongly regular graphs, Johnson graphs, cocktail-party graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees of maximum degree $\le 3$ that are thin trees under some mild assumptions on the parameters of the graph.