FPT-Algorithms for Minimum-Bends Tours

Abstract

This paper discusses the k-Bends Traveling Salesman Problem. In this NP-complete problem, the inputs are n points in the plane and a positive integer k, and we are asked whether we can travel in straight lines through these n points with at most k 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 fixedparameter tractable (FPT). The proof is based on the kernelization approach. We also consider the Rectilinear k-Bends Traveling Salesman Problem, which requires that the line-segments be axis-parallel. a Note that a rectilinear tour with k bends is a cover with k-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.

On the Minimum Link-Length Rectilinear Spanning Path Problem: Complexity and Algorithms

“…The time complexity of our algorithm is , significantly improving the previous known upper bound by Estivill-Castro et al [10]. Our results and techniques can be extended to obtain improved results for the rectilinear Traveling Salesman problem in Euclidean space R ( -RTSP).…”

“…an algorithm of running time for the Constrained 2-RTSP problem, improving the previous algorithm of running time for the problem by Estivill-Castro et al[10].…”

“…Thus, the problem 2-RSP is NP-complete. ◽ A restricted version of the -RSP problem has been studied in the literature [9], [10], in which it is required that each linesegment of the spanning path for the point set must cover all points in that share the same line with . The problem is formally defined as follows.…”

“…Estivill-Castro et al [10] proposed an algorithm for the -RSP problem and claimed that the problem is fixed-parameter tractable. However, Jiang [16] pointed out that the claim was incorrect.…”

Applications of incidence bounds in point covering problems

“…In the Line Cover problem a set of points in R 2 is given and the task is to cover them using either the minimum number of lines, or at most k lines where k is given as a parameter in the input. It is related to Minimum Bend Euclidean TSP and has been studied in connection with facility location problems [7,16]. The Line Cover problem is one of the few low-dimensional geometric problems that are known to be NP-complete [16].…”