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 crossing-free 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 2((1+o(1))root n log n) time while that for counting crossing-free spanning trees runs in O*(7.125(n)) time. The fastest known, nontrivial approximation algorithms for the number of triangulations of S and the number of crossing-free spanning trees of S, respectively, run in time subexponential in n. We present the first non-trivial approximation algorithms for these numbers running in quasi polynomial time. They yield the first quasi-polynomial approximation schemes for the base of the number of triangulations of S and the base of the number of crossing-free spanning trees on S, respectively. (C) 2017 Elsevier B.V. All rights reserved.