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.