# An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

## TL;DR

This paper introduces the first polynomial-time approximation scheme for computing near-optimal high-clearance paths amid polygonal obstacles, balancing path length and obstacle clearance.

## Contribution

It presents a novel polynomial-time approximation scheme for a complex path-planning problem involving obstacle clearance in 2D environments.

## Key findings

- Algorithm computes paths within (1+ε) of optimal
- Time complexity is O(n^2/ε^2 log(n/ε))
- First polynomial-time approximation scheme for this problem

## Abstract

We study a path-planning problem amid a set $\mathcal{O}$ of obstacles in $\mathbb{R}^2$, in which we wish to compute a short path between two points while also maintaining a high clearance from $\mathcal{O}$; the clearance of a point is its distance from a nearest obstacle in $\mathcal{O}$. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let $n$ be the total number of obstacle vertices and let $\varepsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{\varepsilon ^2} \log \frac{n}{\varepsilon})$ a path of total cost at most $(1+\varepsilon)$ times the cost of the optimal path.