Art gallery theorems and algorithms
Art gallery theorems and algorithms
Shortest watchman routes in simple polygons
Discrete & Computational Geometry
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
Approximation algorithms for lawn mowing and milling
Computational Geometry: Theory and Applications
Exploring an Unknown Polygonal Environment with Bounded Visibility
ICCS '01 Proceedings of the International Conference on Computational Sciences-Part I
Approximation algorithms for TSP with neighborhoods in the plane
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
Minimum-cost coverage of point sets by disks
Proceedings of the twenty-second annual symposium on Computational geometry
Guarding galleries and terrains
Information Processing Letters
A PTAS for TSP with neighborhoods among fat regions in the plane
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Polygon exploration with time-discrete vision
Computational Geometry: Theory and Applications
Minimum Covering with Travel Cost
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
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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驴) 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 驴 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.