Rectilinear shortest path and rectilinear minimum spanning tree with neighborhoods

Abstract

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 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. © 2014 Springer International Publishing.