# Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

Yann Disser, Matús Mihalák, Sandro Montanari, Peter Widmayer

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingAcademicpeer-review

## Abstract

We consider a setting where we are given a graph \mathcal {g}=(\mathcal {r},e)\mathcal {g}=(\mathcal {r},e), where \mathcal {r}=\{r_1,\ldots ,r_n\}\mathcal {r}=\{r_1,\ldots ,r_n\} is a set of polygonal regions in the plane. Placing a point p_ip_i inside each region r_ir_i turns gg into an edge-weighted graph g_{\varvec{p}}g_{\varvec{p}}, {\varvec{p}}=\{p_1,\ldots ,p_n\}{\varvec{p}}=\{p_1,\ldots ,p_n\}, where the cost of (r_i,r_j)\in e(r_i,r_j)\in e is the distance between p_ip_i and p_jp_j. The shortest path problem with neighborhoods asks, for given r_sr_s and r_tr_t, to find a placement \varvec{p}\varvec{p} such that the cost of a resulting shortest stst-path in \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}} is minimum among all graphs \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}}. The minimum spanning tree problem with neighborhoods asks to find a placement \varvec{p}\varvec{p} such that the cost of a resulting minimum spanning tree is minimum among all graphs \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}}. We study these problems in the l_1l_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}\mathsf {apx}-hard, even if the neighborhood regions are segments.
Original language English Proc. Third International Symposium on Combinatorial Optimization (ISCO) Springer 208-220 13 https://doi.org/10.1007/978-3-319-09174-7_18 Published - 2014 Yes

### Publication series

Series Lecture Notes in Computer Science 8596