On the complexity of minimum-link path problems

TL;DR

This paper analyzes the computational complexity of minimum-link path problems in various dimensions and settings, providing new hardness results, approximation schemes, and a complete characterization in 2D.

Contribution

It establishes NP-hardness for minimum-link paths in 2D with holes and in 3D terrains, introduces a polynomial-time approximation scheme, and fully characterizes the 2D case.

Findings

Minimum-link diffuse reflection path is NP-hard in 2D with holes.

The problem in 3D terrains is NP-hard but admits a PTAS.

Complete complexity characterization of the problem in 2D.

Abstract

We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several variants of the problem. Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [1] despite a large body of work on the topic. We also resolve the open problem from [2] mentioned in the handbook [3] (see Chapter 27.5, Open problem 3) and The Open Problems Project [4] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.