E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....known. This amounts to constructing a shortest traveling salesperson (TSP) tour on the cells. 252 Christian Icking et al. If the polygonal environment contains obstacles, the problem of finding such a minimum length tour is known to be NP hard [25] and there are some approximation schemes [2,3,18,36]. In a simple polygon without obstacles, the complexity of constructing offline a minimum length tour seems to be open. There are, however, some results concerning the related Hamiltonian cycle and path problems [14,44] and approximations [2,38] 3.1 The Competitive Complexity The following ....

....[25] and there are some approximation schemes [2,3,18,36] In a simple polygon without obstacles, the complexity of constructing offline a minimum length tour seems to be open. There are, however, some results concerning the related Hamiltonian cycle and path problems [14,44] and approximations [2,38]. 3.1 The Competitive Complexity The following result holds true no matter if the robot must return to the start or not. Theorem 2. The competitive complexity of exploring an unknown cellular environment with obstacles equals 2. Proof. Even if we do not know the environment we can apply depth ....

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

....path inside a simple polygon from which each point of the boundary is visible, when start and end points are not specified. In these papers it is always assumed that the range of the robot s visibility is unbounded. Some authors have also studied the case of limited visibility, e.g. Arkin et al. [2] and Ntafos [16] As to the on line version of the polygon exploration problem, Deng et al. 8] were the first to claim that a competitive strategy does exist. In their seminal paper they discussed a subproblem, incurring a competitive factor of 2016. For the rectilinear case, they gave a ....

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

....path inside a simple polygon from which each point of the boundary is visible, when start and end points are not specified. In these papers it is always assumed that the range of the robot s visibility is unbounded. Some authors have also studied the case of limited visibility, e.g. Arkin et al. [3] and Ntafos [22] As to the on line version of the polygon exploration problem, Deng et al. 11] were the first to claim that a competitive strategy does exist. In their seminal paper they discussed a factor of 2016 for a greedy o#ine approach which has to be implemented as an online strategy. ....

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell, Approximation algorithms for lawn mowing and milling, tech. report, Mathematisches Institut, Universitat zu Koln, 1997.

....e. when the environment is already known. This amounts to constructing a shortest traveling salesperson (TSP) tour on the cells. If the polygonal environment contains obstacles, the problem of finding such a minimum length tour is known to be NP hard [8] and there are some approximation schemes [1, 2, 6, 9]. In a simple polygon without obstacles, the complexity of constructing o# line a minimum length tour seems to be open. There are, however, some results concerning the related Hamiltonian cycle and path problems [4, 12] and approximations [1, 11] For our online exploration problem we are not ....

....[8] and there are some approximation schemes [1, 2, 6, 9] In a simple polygon without obstacles, the complexity of constructing o# line a minimum length tour seems to be open. There are, however, some results concerning the related Hamiltonian cycle and path problems [4, 12] and approximations [1, 11]. For our online exploration problem we are not aware of any previous work. 1 ss Figure 1: i) An exploration tour with two obstacles. ii) A shortest TSP tour for the same scene. 2 Preliminary Considerations The first question is if the robot is still able to approximate the optimum ....

E.M.Arkin,S.P.Fekete,andJ.S.B.Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

....smaller connected polygons, each of a given area. This problem, which we call the area partitioning problem, has applications in robotics, where the polygon to be partitioned is a workspace for a number of robots that must perform a terrain covering task such as cleaning, lawn mowing, or milling [1, 10].The workspace is to be partitioned in order to divide the work among the robots. For terrain covering tasks, the goal is to ensure that each point of the workspace is visited at least once. For such tasks, the area of the workspace provides a reasonable estimate for the work to be done. Thus, ....

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

....gives a number of heuristics that match current practice and take into account additional goals of importance in machining. The grid setting for the minimum link Hamiltonian problem is well solved for square grids in [13] All of these problems are discussed in the recent report of Arkin et al. [1]; the survey of Mitchell [14] covers all of this work and more, particularly the various approaches based on approximation algorithms for the traveling salesperson problem. Our problem comes closest to the lawnmower problem, but we may have additional constraints (not every area outside the ....

Arkin, E.M., S.P. Fekete, and J.S.B. Mitchell [1997], "Approximation algorithms for lawnmowing and milling," U. Koln Technical Report ZPR97-255.

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Esther M. Arkin, S'andor P. Fekete, and Joseph S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical Report 97.255, Angewandte Mathematik und Informatik, Universitat zu Koln, 1997. Submitted to Computational Geometry: Theory and Applications.

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Esther M. Arkin, S'andor P. Fekete, and Joseph S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical Report 97.255, Angewandte Mathematik und Informatik, Universitat zu Koln, 1997. Submitted to Computational Geometry: Theory and Applications. 22

....a vast literature on the subject of automatic tool path generation; we refer the reader to Held [21] for a survey and for applications of computational geometry to the problem. The algorithmic study of the problem has focused on the problem of minimizing the length of a milling tour: Arkin et al. [5, 6] show that the problem is NP hard in general. Constant factor approximation algorithms are given in [5, 6, 23] with the best current factor being a 2.5 approximation for min length milling (11 5 approximation for orthogonal simple polygons) For the closely related lawn mowing problem (also known ....

....a survey and for applications of computational geometry to the problem. The algorithmic study of the problem has focused on the problem of minimizing the length of a milling tour: Arkin et al. 5, 6] show that the problem is NP hard in general. Constant factor approximation algorithms are given in [5, 6, 23], with the best current factor being a 2.5 approximation for min length milling (11 5 approximation for orthogonal simple polygons) For the closely related lawn mowing problem (also known as the traveling cameraman problem [23] in which the covering tour is not constrained to stay within P , ....

[Article contains additional citation context not shown here]

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Comput. Geom. Theory Appl., to appear.

....a vast literature on the subject of automatic tool path generation; we refer the reader to Held [21] for a survey and for applications of computational geometry to the problem. The algorithmic study of the problem has focussed on the problem of minimizing the length of a milling tour: Arkin et al. [5, 6] show that the problem is NPhard in general. Constant factor approximation algorithms are given in [5, 6, 23] with the best current factor being a 2.5 approximation for min length milling (11 5approximation for orthogonal simple polygons) For the closely related lawn mowing problem (also known ....

....a survey and for applications of computational geometry to the problem. The algorithmic study of the problem has focussed on the problem of minimizing the length of a milling tour: Arkin et al. 5, 6] show that the problem is NPhard in general. Constant factor approximation algorithms are given in [5, 6, 23], with the best current factor being a 2.5 approximation for min length milling (11 5approximation for orthogonal simple polygons) For the closely related lawn mowing problem (also known as the traveling cameraman problem [23] in which the covering tour is not constrained to stay within P , ....

[Article contains additional citation context not shown here]

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Comput. Geom. Theory Appl., to appear.

....r is equivalent to finding a trajectory that minimises the worst case time for a rendezvous. Being a generalisation of the well known Travelling Salesman Problem, the lawn mower problem is NP hard, so one cannot hope to find an efficient algorithm that computes an optimal tour for any input Y . In [11], Arkin et al. give a number of efficient approximation algorithms for several versions of the problem. However, the scenario of rendezvous search with a known distance d and search radius r is less general, so we do not have to deal with all the difficulties of the lawn mower problem. In fact, ....

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. ZPR Report 97-255, available at ftp://ftp.zpr.uni- koeln.de/pub/paper/zpr97-255.ps.gz, 1997.

....no more than 2L , even if the boundary of P Phi R is disconnnected. This result is also of importance in the more general situation where we have to search a polygonal area A with a scanning device of shape R. This so called lawn mowing problem is NP hard and was considered in [9] and [1]. As it was shown in [1] our inequality allows the construction of approximation algorithms with small approximation factors. 2 The Inequality We will make use of the following proposition, following from Cauchy s formula see [4, 5, 3, 14] Proposition 2.1 (Cauchy 1841) Let C 1 and C 2 be ....

....even if the boundary of P Phi R is disconnnected. This result is also of importance in the more general situation where we have to search a polygonal area A with a scanning device of shape R. This so called lawn mowing problem is NP hard and was considered in [9] and [1] As it was shown in [1], our inequality allows the construction of approximation algorithms with small approximation factors. 2 The Inequality We will make use of the following proposition, following from Cauchy s formula see [4, 5, 3, 14] Proposition 2.1 (Cauchy 1841) Let C 1 and C 2 be closed convex curves of ....

E. Arkin, S. P. Fekete, and J. S. B.Mitchell. Approximation algorithms for lawnmowing and milling. ZPR Report 97-255, available at ftp://ftp.zpr.uni-koeln.de/pub/paper/zpr97-255.ps.gz, 1997

....a Hamiltonian triangulation re nement of the polyhedron. The second method uses rings as building blocks, following a natural ring decomposition from the straight skeleton. We note that these two approaches resemble the two main algorithms for milling a pocket: zigzag and contour machining [4]. The third method uses seamless convex polygons as building blocks, allowing us to optimize the number or length of seams in the overall folding. Acknowledgments. We thank Jin Akiyama for introducing us to the wrapping problem. We thank Anna Lubiw and Ian Munro for valuable comments on this ....

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Tech. Rep. 97.255, Angewandte Mathematik und Informatik, Universitat zu Koln, 1997. Submitted to Computational Geometry: Theory and Applications.

....(possibly disconnected) region R is covered by the disk at some position of the disk. It is easy to see that the problem is NP hard, in general; R may be a set of n well separated points, making the problem that of a TSP with disjoint circular neighborhoods. However, Arkin, Fekete, and Mitchell [24, 25] have shown that the problem is NP hard, even if R is simply connected (e.g. a simple polygon) Their proof also applies to the milling problem, which adds the constraint that the cutter stay within R, in a multiply connected region. What is not yet known, though, is if the milling problem is ....

....and Lenhart [376] who have shown that one can determine Hamiltonicity of a solid grid graph (a grid graph induced by the points that lie inside a simply connected region) in polynomial time. Approximation algorithms for the lawnmowing problem allow one to get within a constant factor of optimal [24, 25, 224]; the best current factor is (3 ffl) based on the algorithm of Arkin, Fekete, and Mitchell [25] together with recent PTAS results for TSP. They also give a 2.5 approximation algorithm for the milling problem; the approximation factor becomes 11 5 if R is a rectilinear simple polygon. The model ....

[Article contains additional citation context not shown here]

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

No context found.

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

No context found.

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universitat zu Koln, 1997.

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Esther M. Arkin, Sandor P. Fekete, and Joseph S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Computational Geometry, 17(1-2):25--50, 2000. 6

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