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Discrete & Computational Geometry
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SCG '90 Proceedings of the sixth annual symposium on Computational geometry
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SIAM Journal on Computing
The exact fitting problem in higher dimensions
Computational Geometry: Theory and Applications
The convex-hull-and-k-line travelling salesman problem
Information Processing Letters
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Discrete Applied Mathematics
Approximation algorithms for lawn mowing and milling
Computational Geometry: Theory and Applications
On Bends and Distances of Paths Among Obstacles in Two-Layer Interconnection Model
IEEE Transactions on Computers
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Discrete & Computational Geometry
Optimal Covering Tours with Turn Costs
SIAM Journal on Computing
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
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Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics)
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Covering a set of points with a minimum number of lines
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
Parameterized Complexity of Geometric Problems
The Computer Journal
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We present efficient algorithms to solve the Line Cover Problem exactly. In this NP-complete problem, the inputs are n points in the plane and a positive integer k, and we are asked to answer if we can cover these n points with at most k lines. Our approach is based on fixed-parameter tractability and, in particular, kernelization. We propose several reduction rules to transform instances of Line Cover into equivalent smaller instances. Once instances are no longer susceptible to these reduction rules, we obtain a problem kernel whose size is bounded by a polynomial function of the parameter k and does not depend on the size n of the input. Our algorithms provide exact solutions and are easy to implement. We also describe the design of algorithms to solve the corresponding optimization problem exactly. We experimentally evaluated ten variants of the algorithms to determine the impact and trade-offs of several reduction rules. We show that our approach provides tractability for a larger range of values of the parameter and larger inputs, improving the execution time by several orders of magnitude with respect to earlier algorithms that use less rules.