Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

We consider a setting where we are given a graph G = (R, E), where R = {R1, . . . , Rn} is a set of polygonal regions in the plane. Placing a point pi inside each region Ri turns G into an edge-weighted graph Gp , p = {p1, . . . , pn}, where the cost of (Ri, Rj) ∈ E is the distance between pi and pj . The Shortest Path Problem with Neighborhoods asks, for given Rs and Rt, to find a placement p such that the cost of a resulting shortest st-path in Gp is minimum among all graphs Gp . The Minimum Spanning Tree Problem with Neighborhoods asks to find a placement p such that the cost of a resulting minimum spanning tree is minimum among all graphs Gp . We study these problems in the L1 metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is APX-hard, even if the neighborhood regions