# Simplification of Polyline Bundles

## TL;DR

This paper studies the problem of simplifying multiple shared polylines simultaneously, proving NP-hardness of approximation, and providing efficient bi-criteria approximation algorithms with fixed-parameter tractability results.

## Contribution

It introduces a generalized polyline bundle simplification problem, proves its NP-hardness, and offers approximation algorithms with bounds based on error tolerance and shared bend points.

## Key findings

- NP-hard to approximate within n^{1/3 - ε} for any ε > 0
- Bi-criteria approximation with O(log(ℓ + n)) factor when allowing δ to be exceeded
- Fixed-parameter tractability in the number of shared bend points

## Abstract

We propose and study a generalization to the well-known problem of polyline simplification. Instead of a single polyline, we are given a set of $\ell$ polylines possibly sharing some line segments and bend points. Our goal is to minimize the number of bend points in the simplified bundle with respect to some error tolerance $\delta$ (measuring Fr\'echet distance) but under the additional constraint that shared parts have to be simplified consistently. We show that polyline bundle simplification is NP-hard to approximate within a factor $n^{1/3 - \varepsilon}$ for any $\varepsilon > 0$ where $n$ is the number of bend points in the polyline bundle. This inapproximability even applies to instances with only $\ell=2$ polylines. However, we identify the sensitivity of the solution to the choice of $\delta$ as a reason for this strong inapproximability. In particular, we prove that if we allow $\delta$ to be exceeded by a factor of $2$ in our solution, we can find a simplified polyline bundle with no more than $O(\log (\ell + n)) \cdot OPT$ bend points in polytime, providing us with an efficient bi-criteria approximation. As a further result, we show fixed-parameter tractability in the number of shared bend points.