Touring Convex Polygons in Polygonal Domain Fences

In the touring polygons problem (TPP), for a given sequence \((s=P_0,P_1,\dots ,P_k,t=P_{k+1})\) of polygons in the plane, where s and t are two points, the goal is to find a shortest path that starts from s, visits each of the polygons in order and ends at t. In the constrained version of TPP, there is another sequence \((F_{0},\dots ,F_{k})\) of polygons called fences, and the portion of the path from \(P_i\) to \(P_{i+1}\) must lie inside the fence \(F_{i}\). TPP is NP-hard for disjoint non-convex polygons, while TPP and constrained TPP are polynomially solvable when the polygons are convex and the fences are simple polygons. In this work, we present the first polynomial time algorithm for solving constrained TPP when the fences are polygonal domains (polygons with holes). Since, the safari problem is a special case of TPP, our algorithm can be used for solving safari problem inside polygons with holes.