Is It FPT to Cover Points with Tours on Minimum Number of Bends (Errata)?

Recent communication by Minghui Jiang has brought to my attention that I overlooked faults in the arguments built while collaborating closely with my PhD student Apichat Heednacram and his co-supervisor Francis Suraweera. These errors unfortunately also escaped the scrutiny of peer-reviews and the formal process of examination. Some results in Apichat's dissertation that were published in this journal (and other outlets) are actually incorrect. In particular, we had reported an FPT-algorithms for the k-Bends Traveling Salesman Problem in ℜ2 and some variants that result from adding constraints to the line-segments that constitute the tour. While the reduction rules to kernelize the problem produce reduced instances, a solution of the kernel instance does not lead directly to a solution of the original instance.

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FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195911003615 ◽

2011 ◽

Vol 21 (02) ◽

pp. 189-213 ◽

Cited By ~ 4

Author(s):

VLADIMIR ESTIVILL-CASTRO ◽

APICHAT HEEDNACRAM ◽

FRANCIS SURAWEERA

Keyword(s):

Traveling Salesman Problem ◽

Traveling Salesman ◽

Fixed Parameter Tractable ◽

Line Segments ◽

Fpt Algorithms ◽

Fixed Parameter ◽

Axis Parallel ◽

Np Complete ◽

Number Of Bends ◽

Straight Lines

This paper discusses the κ-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer κ, and we are asked whether we can travel in straight lines through these n points with at most κ 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 κ-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel. 1 Note that a rectilinear tour with κ bends is a cover with κ-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.

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THE DUBINS TRAVELING SALESMAN PROBLEM WITH CONSTRAINED COLLECTING MANEUVERS

Acta Polytechnica CTU Proceedings ◽

10.14311/app.2016.6.0034 ◽

2016 ◽

Vol 6 ◽

pp. 34

Author(s):

Petr Váňa ◽

Jan Faigl

Keyword(s):

Data Collection ◽

Unmanned Aerial Vehicles ◽

Traveling Salesman Problem ◽

Problem Formulation ◽

Traveling Salesman ◽

Line Segments ◽

Aerial Vehicles ◽

Straight Line ◽

Dubins Traveling Salesman Problem

In this paper, we introduce a variant of the Dubins traveling salesman problem (DTSP) that is called the Dubins traveling salesman problem with constrained collecting maneuvers (DTSP-CM). In contrast to the ordinary formulation of the DTSP, in the proposed DTSP-CM, the vehicle is requested to visit each target by specified collecting maneuver to accomplish the mission. The proposed problem formulation is motivated by scenarios with unmanned aerial vehicles where particular maneuvers are necessary for accomplishing the mission, such as object dropping or data collection with sensor sensitive to changes in vehicle heading. We consider existing methods for the DTSP and propose its modifications to use these methods to address a variant of the introduced DTSP-CM, where the collecting maneuvers are constrained to straight line segments.

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A Model for Multiple Traveling Salesman Problem with Mixed Type Vehicles

Korean Journal of Logistics ◽

10.15735/kls.2018.26.4.008 ◽

2018 ◽

Vol 26 (4) ◽

pp. 115-130

Author(s):

Jin-Wook Kim ◽

채준재

Keyword(s):

Traveling Salesman Problem ◽

Mixed Type ◽

Traveling Salesman ◽

Multiple Traveling Salesman Problem

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Grafos hamiltonianos aplicado al turismo de Panamá

Investigación y Pensamiento Crítico ◽

10.37387/ipc.v7i1.12 ◽

2019 ◽

Vol 7 (1) ◽

pp. 109-113

Author(s):

Julio Trujillo

Keyword(s):

Traveling Salesman Problem ◽

Traveling Salesman

Un problema clásico de Teoría de Grafos es encontrar un camino que pase por varios puntos, sólo una vez, empezando y terminando en un lugar (camino hamiltoniano). Al agregar la condición de que sea la ruta más corta, el problema se convierte uno de tipo TSP (Traveling Salesman Problem). En este trabajo nos centraremos en un problema de tour turístico por la ciudad de Panamá, transformándolo a un problema de grafo de tal manera que represente la situación planteada.

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