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 \(c^n\), where the base c lies between 2.43 and 30. Similarly, the number of spanning trees on S is known to be \(d^n\), where the base d lies between 6.75 and 141.07. The fastest known algorithm for counting triangulations of S runs in \(O^*(2^n)\) time while that for counting spanning trees runs in \(O^*(7.125^n)\) 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.