A QPTAS for the base of the number of crossing-free structures on a planar point set

The number of triangulations of a planar n point set S is known to be c, where the base c lies between 8.65 and 30. Similarly, the number of spanning trees on S is known to be d, where the base d lies between 12.52 and 141.07. The fastest known algorithm for counting triangulations of S runs in O∗(2n) time while that for counting spanning trees on S enumerates them in Ω(12.52) time. The fastest known arbitrarily close approximation algorithms for the base of the number of triangulations of S and the base of the number of spanning trees of S, respectively, run in time subexponential in n. We present the first quasi-polynomial approximation schemes for the base of the number of triangulations of S and the base of the number of spanning trees on S, respectively.