Visiting All Sites with Your Dog

TL;DR

This paper investigates the computational complexity of finding a polygonal curve visiting all points in a set with a bounded Fréchet distance to a given polygonal curve, proving NP-Completeness and providing a polynomial algorithm for convex cases.

Contribution

It establishes NP-Completeness for the general problem and offers a polynomial-time solution for cases where the reference curve is a convex polygon.

Findings

The general problem is NP-Complete.

Polynomial algorithm exists for convex polygon cases.

The problem involves visiting all points with bounded Fréchet distance.

Abstract

Given a polygonal curve P, a pointset S, and an \epsilon > 0, we study the problem of finding a polygonal curve Q whose vertices are from S and has a Frechet distance less or equal to \epsilon to curve P. In this problem, Q must visit every point in S and we are allowed to reuse points of pointset in building Q. First, we show that this problem in NP-Complete. Then, we present a polynomial time algorithm for a special cases of this problem, when P is a convex polygon.