Polygon-Constrained Motion Planning Problems

  • Davide Bilò
  • Yann Disser
  • Luciano Gualà
  • Matúš Mihal’ák
  • Guido Proietti
  • Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8243)

Abstract

We consider the following class of polygon-constrained motion planning problems: Given a set of \(k\) centrally controlled mobile agents (say pebbles) initially sitting on the vertices of an \(n\)-vertex simple polygon \(P\), we study how to plan their vertex-to-vertex motion in order to reach with a minimum (either maximum or total) movement (either in terms of number of hops or Euclidean distance) a final placement enjoying a given requirement. In particular, we focus on final configurations aiming at establishing some sort of visual connectivity among the pebbles, which in turn allows for wireless and optical intercommunication. Therefore, after analyzing the notable (and computationally tractable) case of gathering the pebbles at a single vertex (i.e., the so-called rendez-vous), we face the problems induced by the requirement that pebbles have eventually to be placed at: (i) a set of vertices that form a connected subgraph of the visibility graph induced by \(P\), say \(G(P)\) (connectivity), and (ii) a set of vertices that form a clique of \(G(P)\) (clique-connectivity). We will show that these two problems are actually hard to approximate, even for the seemingly simpler case in which the hop distance is considered.

References

  1. 1.
    Ahmadian, S., Friggstad, Z., Swamy, C.: Local-search based approximation algorithms for mobile facility location problems. arXiv preprint:1301.4478 (2013)Google Scholar
  2. 2.
    Aronov, B., Fortune, S., Wilfong, G.T.: The furthest-site geodesic Voronoi diagram. Discrete Comput. Geom. 9, 217–255 (1993)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Berman, P., Demaine, E.D., Zadimoghaddam, M.: O(1)-approximations for maximum movement problems. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J. (eds.) RANDOM 2011 and APPROX 2011. LNCS, vol. 6845, pp. 62–74. Springer, Heidelberg (2011)Google Scholar
  4. 4.
    Bilò, D., Gualà, L., Leucci, S., Proietti, G.: Exact and approximate algorithms for movement problems on (special classes of) graphs. In: Moscibroda, T., Rescigno, A.A. (eds.) SIROCCO 2013. LNCS, vol. 8179, pp. 322–333. Springer, Heidelberg (2013)Google Scholar
  5. 5.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6, 485–524 (1991)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Demaine, E.D., Hajiaghayi, M., Mahini, H., Sayedi-Roshkhar, A.S., Oveisgharan, S., Zadimoghaddam, M.: Minimizing movement. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA’07), pp. 258–267 (2007)Google Scholar
  7. 7.
    Friggstad, Z., Salavatipour, M.R.: Minimizing movement in mobile facility location problems. ACM Trans. Algorithms 7(3), article 28, 1–22 (2011)Google Scholar
  8. 8.
    Guibas, L.J., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2, 209–233 (1987)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Hershberger, J.: An optimal visibility graph algorithm for triangulated simple polygons. Algorithmica 4(1), 141–155 (1989)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lin, Y.-W., Skiena, S.S.: Complexity aspects of visibility graphs. Int. J. Comput. Geom. Appl. 5(3), 289–312 (1995)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Prencipe, G., Santoro, N.: Distributed algorithms for autonomous mobile robots. In: Navarro, G., Bertossi, L., Kohayakawa, Y. (eds.) TCS 2006. IFIP, vol. 206, pp. 47–62. Springer, Heidelberg (2006)Google Scholar
  12. 12.
    Suri, S., Vicari, E., Widmayer, P.: Simple robots with minimal sensing: From local visibility to global geometry. Int. J. Robot. Res. 27(9), 1055–1067 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Davide Bilò
    • 1
  • Yann Disser
    • 2
  • Luciano Gualà
    • 3
  • Matúš Mihal’ák
    • 4
  • Guido Proietti
    • 5
    • 6
  • Peter Widmayer
    • 4
  1. 1.Dipartimento di Scienze Umanistiche e SocialiUniversità di SassariSassariItaly
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Dipartimento di Ingegneria dell’ImpresaUniversità di Roma Tor VergataRomeItaly
  4. 4.Institut für Theoretische InformatikETHZürichSwitzerland
  5. 5.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità dell’AquilaCoppito L’AquilaItaly
  6. 6.Istituto di Analisi dei Sistemi ed InformaticaCNRRomaItaly

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