eCOMPASS – TR – 046 Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

We study the geometric shortest path and the minimum spanning tree problem with neighborhoods in the L1 metric. In this setting, 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 an edge is the distance between the points. The Shortest Path Problem with Neighborhoods asks, for given Rs and Rt, to find a placement p such that the resulting shortest s-t path in Gp is smallest among all graphs Gp. The Minimum Spanning Tree Problem with Neighborhoods asks for a placement p such that the resulting minimum spanning tree of Gp has the smallest cost among all minimum spanning trees of 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 NP-hard, even if the neighborhood regions