Touring Disjoint Polygons Problem Is NP-Hard

Abstract

In the Touring Polygons Problem (TPP) there is a start point s, a sequence of simple polygons \(\mathcal{P}=(P_1,\dots,P_k)\) and a target point t in the plane. The goal is to obtain a path of minimum possible length that starts from s, visits in order each of the polygons in \(\mathcal{P}\) and ends at t. This problem has a polynomial time algorithm when the polygons in \(\mathcal{P}\) are convex and is NP-hard in general case. But, it has been open whether the problem is NP-hard when the polygons are pairwise disjoint. In this paper, we prove that TPP is also NP-hard when the polygons are pairwise disjoint in any L p norm even if each polygon consists of at most two line segments. This result solves an open problem from STOC ’03 and complements recent approximation results.