Map Matching with Simplicity Constraints

TL;DR

This paper investigates the computational complexity of map matching with simplicity constraints, proving NP-completeness for finding simple cycles close to a given curve under the Fréchet distance.

Contribution

It establishes the NP-completeness of determining simple cycles in planar graphs within a specified Fréchet distance to a simple closed curve, highlighting computational limitations.

Findings

NP-completeness of the problem

Implications for variants of map matching

Challenges in guaranteeing simple solutions

Abstract

We study a map matching problem, the task of finding in an embedded graph a path that has low distance to a given curve in R^2. The Fr\'echet distance is a common measure for this problem. Efficient methods exist to compute the best path according to this measure. However, these methods cannot guarantee that the result is simple (i.e. it does not intersect itself) even if the given curve is simple. In this paper, we prove that it is in fact NP-complete to determine the existence a simple cycle in a planar straight-line embedding of a graph that has at most a given Fr\'echet distance to a given simple closed curve. We also consider the implications of our proof on some variants of the problem.