# Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

## TL;DR

This paper proves polynomial bounds on the number of non-crossing Hamiltonian paths and cycles in planar point sets, enabling efficient listing and approximation algorithms based on geometric parameters.

## Contribution

It establishes polynomial bounds relating non-crossing Hamiltonian structures to geometric parameters, allowing output-polynomial listing and approximation algorithms.

## Key findings

- Number of non-crossing Hamiltonian paths is polynomially bounded by non-crossing paths.
- Number of non-crossing Hamiltonian cycles is polynomially bounded by surrounding cycles.
- Provides polynomial-time approximation algorithms for counting these structures.

## Abstract

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.