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### Euclidean TSP in Narrow Strip [article]

Henk Alkema and Mark de Berg and Sándor Kisfaludi-Bak
2020 arXiv   pre-print
We investigate how the complexity of Euclidean TSP for point sets P inside the strip (-∞,+∞)× [0,δ] depends on the strip width δ. We obtain two main results.  ...  First, for the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in O(nlog^2 n) time using an existing algorithm) is guaranteed to  ...  In particular, we assume that the point set P is contained in the strip (−∞, ∞) × [0, δ] for some relatively small δ and investigate how the complexity of Euclidean TSP depends on the parameter δ.  ...

### Euclidean TSP in Narrow Strips

Henk Alkema, Mark de Berg, Sándor Kisfaludi-Bak, Danny Z. Chen, Sergio Cabello
2020 International Symposium on Computational Geometry
We investigate how the complexity of {Euclidean TSP} for point sets P inside the strip (-∞,+∞)×[0,δ] depends on the strip width δ.  ...  . - For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in O(n log²n) time using an existing algorithm) is guaranteed to be a  ...  Euclidean TSP in Narrow Strips Recall that each edge e ∈ F is obtained from the corresponding edge e ∈ F by moving one or both endpoints along the edge e itself.  ...

### Watchman tours for polygons with holes

2012 Computational geometry
A watchman tour in a polygonal domain (for short, polygon) is a closed curve in the polygon such that every point in the polygon is visible from at least one point of the tour.  ...  Apart from the multiplicative constant, this bound is tight in the worst case. We then generalize our result to watchman tours in polyhedra with holes in 3-space.  ...  The authors are grateful to two anonymous reviewers for uncovering gaps in our initial formulation of Theorem 2 and in the NP-hardness proof.  ...

### An asymptotic approximation of the traveling salesman problem with uniform non-overlapping time windows

Omar Rifki, Thierry Garaix, Christine Solnon
2021 2021 IEEE 17th International Conference on Automation Science and Engineering (CASE)
We develop a continuous asymptotic approximation of the traveling salesman problem with time windows in the Euclidean plane, constructing upon the well-known Beardwood-Halton-Hemmersley theorem.  ...  Computational experiments on random TSP with time windows instances show that the proposed asymptotic approximations of tour lengths and arrival times are close to the actual optimal values.  ...  Noting that the BHH formula underestimates tour lengths in elongated areas, Daganzo  proposed a strip strategy method, which efficiently computes the optimal tour length in those types of areas.  ...

### Traveling Salesman Problem Based Auto-Router for Designing LEDs Applications with Conductive Inkjet Printing

Tung D. TA, Fuminori OKUYA, Yoshihiro KAWAHARA
2018 SICE Journal of Control Measurement and System Integration
It is possible to try existing auto-routers in computer aided design tools to automatically route these LEDs.  ...  In this paper, we propose an LED auto-router which computationally generates a conductive pattern to balance brightness of multiple LEDs without the need of additional resistors.  ...  Fig. 10 10 Conversion of an excessively narrow conductive strip to its equivalent meander line.  ...

### Approximation algorithms for lawn mowing and milling☆☆A preliminary version of this paper was entitled "The lawnmower problem" and appears in the Proc. 5th Canad. Conf. Comput. Geom., Waterloo, Canada, 1993, pp. 461–466

Esther M. Arkin, Sándor P. Fekete, Joseph S.B. Mitchell
2000 Computational geometry
Furthermore, we give a simple 6 5 -approximation algorithm for the TSP problem in simple grid graphs, which leads to an 11 5 -approximation algorithm for milling simple rectilinear polygons.  ...  The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general.  ...  (These problems are clearly NP-hard, from the fact that the Euclidean TSP is NP-hard.)  ...

### Points on Computable Curves

Xiaoyang Gu, Jack Lutz, Elvira Mayordomo
2006 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
In this paper we characterize those points of Euclidean space that lie on computable curves of finite length.  ...  The "analyst's traveling salesman theorem" of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length.  ...  Note that if all these points involved lie in a very narrow strip, it is guaranteed that the newly added line segments are very close to the longer line segment they replace.  ...

### Design and Analysis of Randomized and Approximation Algorithms (Dagstuhl Seminar 11241)

Martin Dyer, Uriel Feige, Alan M. Frieze, Marek Karpinski, Marc Herbstritt
2011 Dagstuhl Reports
Often, these two notions go hand-in-hand. Acknowlegement. We thank Annette Beyer and Angelika Mueller-von Brochowski for their continuous support and help in organizing this workshop.  ...  Nevertheless, practical necessity dictates that acceptable solutions be found in a reasonable time.  ...  In this paper we present an approximation algorithm for the strip packing problem with approximation ratio of 5/3 + for any > 0.  ...

### EFFICIENT TOUR PLANNING FOR A MEASUREMENT VEHICLE BY COMBINING NEXT BEST VIEW AND TRAVELING SALESMAN

J. Gehrung, M. Hebel, M. Arens, U. Stilla
2021 The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences
In this paper, we formulate the basic problem, discuss it in context of the existing literature and present an iterative solution algorithm.  ...  Combining both problems results in a new variation of the Traveling Salesman Problem, which we refer to as the Explorational Traveling Salesman Problem.  ...  Arc costs are initialized with the Euclidean distance.  ...

### Minimal link visibility paths inside a simple polygon

Muhammed H. Alsuwaiyel, D.T. Lee
1993 Computational geometry
Specifically, given a set S of points (edges) in P, the problems of finding a tour with a minimum number of turns that visits each point (edge) in S exactly once are also shown to be NP-hard.  ...  In proving this main result, we show two other related problems to be NP-hard as well.  ...  As shown in Fig. 8a and 8b, P is identical to P' except for the addition of two polygonal narrow strips of n turns each, where n is the number of edges in P'.  ...

### Approximation algorithms for TSP with neighborhoods in the plane

2003 Journal of Algorithms
In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region.  ...  As a generalization of the classical Euclidean TSP, TSPN is also NP-hard.  ...  the case of the Euclidean TSP on points, as in  .  ...

### Rectilinear Steiner Trees in Narrow Strips [article]

Henk Alkema, Mark de Berg
2021 arXiv   pre-print
A rectilinear Steiner tree for a set P of points in ℝ^2 is a tree that connects the points in P using horizontal and vertical line segments.  ...  We investigate how the complexity of Minimal Rectilinear Steiner Tree for point sets P inside the strip (-∞,+∞)× [0,δ] depends on the strip width δ.  ...  If the point set in P is "almost 1-dimensional" in the sense that the points lie in a narrow strip R × [0, δ], then can we solve Minimum Rectilinear Steiner Tree more efficiently than in the general case  ...

### Matrix Encoding Networks for Neural Combinatorial Optimization [article]

Yeong-Dae Kwon, Jinho Choo, Iljoo Yoon, Minah Park, Duwon Park, Youngjune Gwon
2021 arXiv   pre-print
In this paper, we introduce Matrix Encoding Network (MatNet) and show how conveniently it takes in and processes parameters of such complex CO problems.  ...  Many CO problems of practical importance can be specified in a matrix form of parameters quantifying the relationship between two groups of items.  ...  We thank Kevin Tierney for reviewing the MIP methods used in our experiments and giving us helpful tips and comments. We declare no third party funding or support.  ...

### Evolving Combinatorial Problem Instances That Are Difficult to Solve

Jano I. van Hemert
2006 Evolutionary Computation
Problem instances acquired through this technique are more difficult than ones found in popular benchmarks.  ...  We analyse these evolved instances with the aim to explain their difficulty in terms of structural properties, thereby exposing the weaknesses of corresponding algorithms.  ...  Figure 13 : 13 Two typical examples of the Euclidean TSP problem, both clustered usingGDBSCAN (eps=41,m=5).  ...

### Performance and Design of Mobility Allowance Shuttle Transit Services: Bounds on the Maximum Longitudinal Velocity

Luca Quadrifoglio, Randolph W. Hall, Maged M. Dessouky
2006 Transportation Science
The resulting narrow gap between them under realistic operating conditions allows us to evaluate the service in terms of velocity and capacity versus demand.  ...  The relationships obtained can be helpful in the design of MAST systems to set the main parameters of the service, such as slack time and headway.  ...  Acknowledgments The research reported in this paper was partially supported by the National Science Foundation under Grant NSF/USDOT-0231665.  ...
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