Embedding Ray Intersection Graphs and Global Curve Simplification

TL;DR

This paper demonstrates a polynomial embedding of circle graphs as ray intersection graphs and proves that the global curve simplification problem under directed Hausdorff distance is NP-hard, highlighting computational complexity challenges.

Contribution

It introduces a polynomial-size embedding of circle graphs as ray intersection graphs and establishes NP-hardness of the global curve simplification problem.

Findings

Circle graphs can be embedded as intersection graphs of rays with polynomial complexity.

Global curve simplification under directed Hausdorff distance is NP-hard.

Provides a new perspective on the complexity of curve simplification problems.

Abstract

We prove that circle graphs (intersection graphs of circle chords) can be embedded as intersection graphs of rays in the plane with polynomial-size bit complexity. We use this embedding to show that the global curve simplification problem for the directed Hausdorff distance is NP-hard. In this problem, we are given a polygonal curve $P$ and the goal is to find a second polygonal curve $P'$ such that the directed Hausdorff distance from $P'$ to $P$ is at most a given constant, and the complexity of $P'$ is as small as possible.