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Minimum Covering with Travel Cost
- Publication Year :
- 2011
-
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 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_\infty 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(r/a}+(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.<br />17 pages, 12 figures; extended abstract appears in ISAAC 2009, full version to appear in Journal of Combinatorial Optimization
- Subjects :
- Computational Geometry (cs.CG)
FOS: Computer and information sciences
Control and Optimization
0102 computer and information sciences
01 natural sciences
Travelling salesman problem
Combinatorics
Path length
Computer Science - Data Structures and Algorithms
Range (statistics)
Discrete Mathematics and Combinatorics
Point (geometry)
Data Structures and Algorithms (cs.DS)
0101 mathematics
Linear combination
Mathematics
Applied Mathematics
010102 general mathematics
Visibility (geometry)
Approximation algorithm
Computer Science Applications
Computational Theory and Mathematics
010201 computation theory & mathematics
Polygon
Computer Science - Computational Geometry
F.2.2
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....dada9430362aca716e572e2e8fa1376c