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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Copyright information

© 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

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