A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting
MohammadHossein Bateni, Erik D. Demaine, MohammadTaghi Hajiaghayi, Dániel Marx · 2016
We present the first polynomial-time approximation scheme (PTAS), i.e., (1+ε)-approximation algorithm for any constant ε> 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn (loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing light-weight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems.