FPT-Algorithms for Minimum-Bends Tours
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This paper discusses the ?-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer ?, and we are asked whether we can travel in straight lines through these n points with at most ? bends. There are a number of applications where minimizing the number of bends in the tour is desirable because bends are considered very costly. We prove that this problem is fixed-parameter tractable (FPT). The proof is based on the kernelization approach. We also consider the RECTILINEAR ?-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel. [note: An earlier version of the Rectilinear ?-Bends Traveling Salesman Problem has been published in COCOON 2010.]1 Note that a rectilinear tour with ? bends is a cover with ?-line segments, and therefore a cover by lines. We introduce two types of constraints derived from the distinction between line-segments and lines. We derive FPT-algorithms with different techniques and improved time complexity for these cases.
International Journal of Computational Geometry and Applications (IJCGA)
Electronic version of an article published in International Journal of Computational Geometry and Applications (IJCGA), Vol. 21(2), 2011, pp. 189-213, http://dx.doi.org/10.1142/S0218195911003615. Copyright World Scientific Publishing Company http://www.worldscinet.com/ijcga/ijcga.shtml
Analysis of Algorithms and Complexity