A heuristic algorithm for straight-line embedding of a hamiltonian cycle onto a given set of points inside simple polygons

TL;DR

This paper presents a heuristic algorithm for embedding a Hamiltonian cycle as a straight-line, planar cycle inside a simple polygon with a set of points, addressing a complex geometric problem with practical heuristic solutions.

Contribution

It introduces a novel heuristic algorithm with specific complexity bounds for embedding Hamiltonian cycles in polygons, advancing the approach to a challenging geometric problem.

Findings

The heuristic algorithm successfully embeds Hamiltonian cycles in polygons.

The algorithm operates with a time complexity of O(r(n^2m + n^3)).

The space complexity of the algorithm is O(n^2 + m).

Abstract

This paper investigated the problem of embedding a simple Hamiltonian Cycle with n vertices on n points inside a simple polygon. This problem seeks to embed a straight-line cycle (without bends), which does not intersect either itself or the sides of the polygon, i.e., it is planar. This problem is a special case of an open problem to find a simple Hamiltonian (s, X, t)-path (a simple path that starts at s and ends at t, where s, t, and all other vertices within the path are a member of set X) inside a simple polygon, which does not intersect itself or the sides of the polygon. The complexity of the problem in this paper is not verified yet, and it is an open problem. However, similar problems are resolved that are NP-Complete. A heuristic algorithm with time complexity of O(r(n2m + n3)) and space complexity of O(n2 + m) is proposed to solve the problem.