Skip to main content

Minimum covering with travel cost

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 show that this problem is NP-hard, even for the special cases of rectilinear polygons and L scan range 1, and negligible small travel cost or negligible travel cost. For rectilinear MWPDV milling in grid polygons we present a 2.5-approximation 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.

This is a preview of subscription content, access via your institution.

References

  • Alt H, Arkin EM, Brönnimann H, Erickson J, Fekete SP, Knauer C, Lenchner J, Mitchell JSB, Whittlesey K (2006) Minimum-cost coverage of point sets by disks. In: Proc. 22nd ACM symposium on computational geometry, pp 449–458

  • Amit Y, Mitchell JSB, Packer E (2010) Locating guards for visibility coverage of polygons. Int J Comput Geom Appl (to appear)

  • Arkin EM, Fekete SP, Mitchell JSB (2000) Approximation algorithms for lawn mowing and milling. Comput Geom Theory Appl 17(1–2):25–50

    MathSciNet  MATH  Article  Google Scholar 

  • Baumgartner T, Fekete SP, Kröller A, Schmidt C (2010) Exact solutions and bounds for general art gallery problems. In: Proc. SIAM-ACM workshop on algorithm engineering and experiments (ALENEX 2010)

  • Baur C, Fekete SP (2001) Approximation of geometric dispersion problems. Algorithmica 30(3):451–470

    MathSciNet  MATH  Article  Google Scholar 

  • Bhattacharya A, Ghosh SK, Sarkar S (2001) Exploring an unknown polygonal environment with bounded visibility. In: International conference on computational science (1). LNCS, vol 2073. Springer, Berlin, pp 640–648

    Google Scholar 

  • Chin W-P, Ntafos S (1988) Optimum watchman routes. In: Proc 2nd ACM Symposium on Computational Geometry, vol. 28(1), pp 39–44

  • Chin W-P, Ntafos SC (1991) Shortest watchman routes in simple polygons. Discrete Comput Geom 6:9–31

    MathSciNet  MATH  Article  Google Scholar 

  • Dumitrescu A, Mitchell JSB (2003) Approximation algorithms for TSP with neighborhoods in the plane. J Algorithms 48(1):135–159

    MathSciNet  MATH  Article  Google Scholar 

  • Efrat A, Har-Peled S (2006) Guarding galleries and terrains. Inf Process Lett 100(6):238–245

    MathSciNet  MATH  Article  Google Scholar 

  • Fekete SP, Schmidt C (2008) Polygon exploration with discrete vision. In: CoRR, 0807.2358

  • Fekete SP, Schmidt C (2009) Low-cost tours for nearsighted watchmen with discrete vision. In: 25th European workshop on computational geometry, pp 171–174

  • Fekete SP, Schmidt C (2010) Polygon exploration with time-discrete vision. Comput Geom Theory Appl 43(2):148–168

    MathSciNet  MATH  Article  Google Scholar 

  • Fekete SP, Mitchell JSB, Schmidt C (2009) Minimum covering with travel cost. In: Proc. 20th international symposium on algorithms and computation. Lecture notes in computer science, vol 5878. Springer, Berlin, pp 393–402

    Google Scholar 

  • Hochbaum DS, Maass W (1985) Approximation schemes for covering and packing problems in image processing and vlsi. J ACM 32(1):130–136

    MathSciNet  MATH  Article  Google Scholar 

  • Itai A, Papadimitriou CH, Szwarcfiter JL (1982) Hamilton paths in grid graphs. SIAM J Comput 11(4):676–686

    MathSciNet  MATH  Article  Google Scholar 

  • Kershner R (1939) The number of circles covering a set. Am J Math 61:665–667

    MathSciNet  Article  Google Scholar 

  • Mitchell JSB (1999) Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J Comput 28:1298–1309

    MathSciNet  MATH  Article  Google Scholar 

  • Mitchell JSB (2007) A PTAS for TSP with neighborhoods among fat regions in the plane. In: Proc. 18th annual ACM-SIAM symposium on discrete algorithms, pp 11–18

  • O’Rourke J (1987) Art gallery theorems and algorithms. International series of monographs on computer science. Oxford University Press, New York

    MATH  Google Scholar 

  • Rosenstiehl P, Tarjan RE (1986) Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput Geom 1:343–353

    MathSciNet  MATH  Article  Google Scholar 

  • Tóth LF (1949) Über dichteste Kreislagerung und dünnste Kreisüberdeckung. Comment Math Helv 23(1):342–349

    MathSciNet  MATH  Article  Google Scholar 

  • Wagner IA, Lindenbaum M, Bruckstein AM (2000) MAC vs. PC: Determinism and randomness as complementary approaches to robotic exploration of continuous unknown domains. Int J Robotics Res 19(1):12–31

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christiane Schmidt.

Additional information

A preliminary extended abstract of this paper appears in Fekete et al. (2009). A 4-page abstract based on Sects. 3 and 4 of this paper appeared in the informal and non-selective workshop “EuroCG”, March 2009 (Fekete and Schmidt 2009).

Joseph S.B. Mitchell is partially supported by the National Science Foundation (CCF-0528209, CCF-0729019), Metron Aviation, and NASA Ames.

Christiane Schmidt is supported by DFG Focus Program “Algorithm Engineering” (SPP 1307) project “RoboRithmics” (Fe 407/14-1).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fekete, S.P., Mitchell, J.S.B. & Schmidt, C. Minimum covering with travel cost. J Comb Optim 24, 32–51 (2012). https://doi.org/10.1007/s10878-010-9303-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-010-9303-0

Keywords

  • Covering
  • Minimum Watchman Problem
  • Limited visibility
  • Lawn mowing
  • Bicriteria problems
  • Approximation algorithm
  • PTAS