A QPTAS for the Base of the Number of Triangulations of a Planar Point Set

TL;DR

This paper introduces the first quasi-polynomial approximation scheme for estimating the base of the number of triangulations of a planar point set, improving understanding and computational approaches for this combinatorial problem.

Contribution

It presents the first QPTAS for approximating the base of the number of triangulations in planar point sets, advancing computational geometry techniques.

Findings

Provides a quasi-polynomial time approximation scheme.

Improves the approximation of the base c between 2.43 and 30.

Enhances algorithms for counting triangulations.

Abstract

The number of triangulations of a planar n point set is known to be $c^n$, where the base $c$ lies between $2.43$ and $30.$ The fastest known algorithm for counting triangulations of a planar n point set runs in $O^*(2^n)$ time. The fastest known arbitrarily close approximation algorithm for the base of the number of triangulations of a planar n point set runs in time subexponential in $n.$ We present the first quasi-polynomial approximation scheme for the base of the number of triangulations of a planar point set.