FPT-Algorithms for Minimum-Bends Tours

Abstract

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.