Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

Abstract

We consider a setting where we are given a graph $$\mathcal {G}=(\mathcal {R},E)$$ , where $$\mathcal {R}=\{R_1,\ldots ,R_n\}$$ is a set of polygonal regions in the plane. Placing a point $$p_i$$ inside each region $$R_i$$ turns $$G$$ into an edge-weighted graph $$G_{\varvec{p}}$$ , $${\varvec{p}}=\{p_1,\ldots ,p_n\}$$ , where the cost of $$(R_i,R_j)\in E$$ is the distance between $$p_i$$ and $$p_j$$ . The Shortest Path Problem with Neighborhoods asks, for given $$R_s$$ and $$R_t$$ , to find a placement $$\varvec{p}$$ such that the cost of a resulting shortest $$st$$ -path in $$\mathcal {G}_{\varvec{p}}$$ is minimum among all graphs $$\mathcal {G}_{\varvec{p}}$$ . The Minimum Spanning Tree Problem with Neighborhoods asks to find a placement $$\varvec{p}$$ such that the cost of a resulting minimum spanning tree is minimum among all graphs $$\mathcal {G}_{\varvec{p}}$$ . We study these problems in the $$L_1$$ metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is $$\mathsf {APX}$$ -hard, even if the neighborhood regions are segments.