Minimum Covering with Travel Cost

  • Sándor P. Fekete
  • Joseph S. B. Mitchell
  • Christiane Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)

Abstract

Given a polygon and a visibility range, the Myopic Watchman Problem with Discrete Vision (MWPDV) asks for a closed path P and a set of scan points \({\mathcal S}\), such that (i) every point of the polygon is within visibility range of a scan point; and (ii) path length plus weighted sum of scan number along the tour is minimized. Alternatively, the bicriteria problem (ii’) aims at minimizing both scan number and tour length. We consider both lawn mowing (in which tour and scan points may leave P) and milling (in which tour, scan points and visibility must stay within P) variants for the MWPDV; even for simple special cases, these problems are NP-hard.

We sketch a 2.5-approximation for rectilinear MWPDV milling in grid polygons with unit scan range; this holds for the bicriteria version, thus for any linear combination of travel cost and scan cost. For grid polygons and circular unit scan range, we describe a bicriteria 4-approximation. These results serve as stepping stones for the general case of circular scans with scan radius r and arbitrary polygons of feature size a, for which we extend the underlying ideas to a \(\pi(\frac{r}{a}+\frac{r+1}{2})\) bicriteria approximation algorithm. Finally, we describe approximation schemes for MWPDV lawn mowing and milling of grid polygons, for fixed ratio between scan cost and travel cost.

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References

  1. 1.
    Alt, H., Arkin, E.M., Brönnimann, H., Erickson, J., Fekete, S.P., Knauer, C., Lenchner, J., Mitchell, J.S.B., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Symposium on Computational Geometry, pp. 449–458 (2006)Google Scholar
  2. 2.
    Arkin, E.M., Fekete, S.P., Mitchell, J.S.B.: Approximation algorithms for lawn mowing and milling. Comput. Geom. Theory Appl. 17(1-2), 25–50 (2000)MATHMathSciNetGoogle Scholar
  3. 3.
    Bhattacharya, A., Ghosh, S.K., Sarkar, S.: Exploring an unknown polygonal environment with bounded visibility. In: Alexandrov, V.N., Dongarra, J., Juliano, B.A., Renner, R.S., Tan, C.J.K. (eds.) ICCS-ComputSci 2001. LNCS, vol. 2073, pp. 640–648. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Chin, W.-P., Ntafos, S.: Optimum watchman routes. In: Proc. 2nd ACM Symposium on Computational Geometry, vol. 28(1), pp. 39–44 (1988)Google Scholar
  5. 5.
    Chin, W.-P., Ntafos, S.C.: Shortest watchman routes in simple polygons. Discrete & Computational Geometry 6, 9–31 (1991)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Efrat, A., Har-Peled, S.: Guarding galleries and terrains. Information Processing Letters 100(6), 238–245 (2006)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Fekete, S.P., Schmidt, C.: Polygon exploration with discrete vision. CoRR, abs/0807.2358 (2008)Google Scholar
  9. 9.
    Fekete, S.P., Schmidt, C.: Low-cost tours for nearsighted watchmen with discrete vision. In: 25th European Workshop on Comput.Geom., pp. 171–174 (2009)Google Scholar
  10. 10.
    Fekete, S.P., Schmidt, C.: Polygon exploration with time-discrete vision. Computational Geometry 43(2), 148–168 (2010)MATHCrossRefGoogle Scholar
  11. 11.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and vlsi. J. ACM 32(1), 130–136 (1985)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamilton paths in grid graphs. SIAM Journal on Computing 11(4), 676–686 (1982)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mitchell, J.S.B.: A PTAS for TSP with neighborhoods among fat regions in the plane. In: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 11–18 (2007)Google Scholar
  14. 14.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Internat. Series of Monographs on Computer Science. Oxford University Press, New York (1987)MATHGoogle Scholar
  15. 15.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: MAC vs. PC: Determinism and randomness as complementary approaches to robotic exploration of continuous unknown domains. ROBRES: The International Journal of Robotics Research 19(1), 12–31 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sándor P. Fekete
    • 1
  • Joseph S. B. Mitchell
    • 2
  • Christiane Schmidt
    • 1
  1. 1.Algorithms GroupBraunschweig Institute of TechnologyGermany
  2. 2.Department of Applied Mathematics and StatisticsStony Brook University 

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