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 = {R ₁,..., 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 _p, p = {p ₁,..., p _n), where the cost of (R _i, R _j) ∈ 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 p such that the cost of a resulting shortest st-path in G _p is minimum among all graphs G _p. 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 G _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 APX-hard, even if the neighborhood regions are segments.