Reduction Rules Deliver Efficient FPT Algorithms for Covering Points with Lines
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Author(s)
Estivill-Castro, Vladimir
Heednacram, Apichat
Suraweera, Francis
Year published
2009
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Show full item recordAbstract
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 ...
View more >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 previously known algorithms.
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View more >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 previously known algorithms.
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Journal Title
Journal of Experimental Algorithmics
Volume
14
Copyright Statement
© ACM, 2009. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Journal of Experimental Algorithmics (JEA), Volume 14, December 2009, http://doi.acm.org/10.1145/1498698.1626535
Subject
Analysis of Algorithms and Complexity
Pure Mathematics
Applied Mathematics
Computation Theory and Mathematics