Finding Two Disjoint Simple Paths on Two Sets of Points is NP-Complete

TL;DR

This paper proves that finding two disjoint simple paths on two point sets is NP-Complete, introduces reductions from planar Hamiltonian path, and proposes a heuristic algorithm with O(n^4) complexity.

Contribution

It establishes the NP-Completeness of the problem and presents a heuristic algorithm for practical solutions.

Findings

NP-Completeness proven via reduction from planar Hamiltonian path

Heuristic algorithm with O(n^4) complexity for path finding

Reduction to obstacle path problem also shown

Abstract

Finding two disjoint simple paths on two given sets of points is a geometric problem introduced by Jeff Erickson. This problem has various applications in computational geometry, like robot motion planning, generating polygon etc. We will present a reduction from planar Hamiltonian path to this problem, and prove that it is NP-Complete. To the best of our knowledge, no study has considered its complexity up until now. We also present a reduction from planar Hamiltonian path problem to the problem of finding a path on given points in the presence of arbitrary obstacles and prove that it is NP-Complete too. Also, we present a heuristic algorithm with time complexity of O(n4) to solve this problem. The proposed algorithm first calculates the convex hull for each of the entry points and then produces two simple paths on the two entry point sets