ISCO 2014: Combinatorial Optimization pp 208-220

# Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

• Yann Disser
• Matúš Mihalák
• Sandro Montanari
• Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)

## 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.

### Keywords

Neighborhoods Minimum spanning tree Shortest path

## Notes

### Acknowledgments

This work was supported by the EU FP7/2007-2013 (DG CONNECT.H5-Smart Cities and Sustainability), under grant agreement no. 288094 (project eCOMPASS) and by the Alexander von Humboldt-Foundation.

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Yann Disser
• 1
• Matúš Mihalák
• 2
• Sandro Montanari
• 2
• Peter Widmayer
• 2
1. 1.Department of MathematicsTU BerlinBerlinGermany
2. 2.Department of Computer ScienceETH ZurichZurichSwitzerland