A New Spatial Approach for Efficient Transformation of Equality - Generalized TSP to TSP

The Equality Generalized Travelling Salesman Problem (E-GTSP) asks to find a Hamiltonian cycle visiting each group exactly once, where each group represents a type of visiting node. This can represent a range of combinatorial optimization problem of NP-hard type like planning, logistics, etc. Its solution requires transformation of E-GTSP to TSP before solving it using a given TSP solver. This paper presents 5 different search-algorithms for optimal transformation which considers spatial spread of nodes of each group. Algorithms are tested over 15 cities with different street-network’s fractal-dimension for 5 instances of group-counts each. It’s observed that the R-Search algorithm, which selects nodes from each group depending upon their radial separation with respect to the start-end point, is the optimal search criterion among all other algorithms with a mean length error of 8.8%. This study will help developers and researchers to answer complex routing problems from a spatial perspective. ∗Corresponding author Email address: zia@itu.edu.tr,mohammed.zia33@gmail.com (Mohammed Zia) URL: https://linkedin.com/in/zia33 (Mohammed Zia) Submitted to FOSS4G 2017 Conference Proceedings, Boston, USA. September 20, 2017 FOSS4G 2017 Academic Program Transformation of E-GTSP to TSP

[1]  William Rowan Hamilton.  LVI. Memorandum respecting a new system of roots of unity , 1856 .

[2]  G. Croes.  A Method for Solving Traveling-Salesman Problems , 1958 .

[3]  G. Clarke,et al.  Scheduling of Vehicles from a Central Depot to a Number of Delivery Points , 1964 .

[4]  Jan Karel Lenstra,et al.  Some Simple Applications of the Travelling Salesman Problem , 1975 .

[5]  Daniel J. Rosenkrantz,et al.  An Analysis of Several Heuristics for the Traveling Salesman Problem , 1977, SIAM J. Comput..

[6]  G. Laporte,et al.  Generalized Travelling Salesman Problem Through n Sets Of Nodes: An Integer Programming Approach , 1983 .

[7]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[8]  Gilbert Laporte,et al.  Generalized travelling salesman problem through n sets of nodes: the asymmetrical case , 1987, Discret. Appl. Math..

[9]  Fred Glover,et al.  Tabu Search: A Tutorial , 1990 .

[10]  James C. Bean,et al.  A Lagrangian Based Approach for the Asymmetric Generalized Traveling Salesman Problem , 1991, Oper. Res..

[11]  A. J. Jones,et al.  Estimating the Held-Karp lower bound for the geometric TSP , 1997 .

[12]  Keld Helsgaun,et al.  An effective implementation of the Lin-Kernighan traveling salesman heuristic , 2000, Eur. J. Oper. Res..

[13]  Gregory Gutin,et al.  The Greedy Algorithm for the Symmetric TSP , 2007, Algorithmic Oper. Res..

[14]  Keld Helsgaun,et al.  General k-opt submoves for the Lin–Kernighan TSP heuristic , 2009, Math. Program. Comput..

[15]  Kevin M. Curtin,et al.  A Comparative Analysis of Traveling Salesman Solutions from Geographic Information Systems , 2014, Trans. GIS.

[16]  Christos A. Kontovas,et al.  Dynamic vehicle routing problems: Three decades and counting , 2016, Networks.