A Greedy Reconstruction Heuristic for Solving the Minimum Travelling Salesman Tour Problem

Authors

  • Santosh Kumar OAM Department of Mathematical and Geospatial Sciences, School of Sciences, RMIT University, Melbourne Australia
  • Elias Munapo School of Economics and Decision Sciences, North West University, Mafikeng 2735, South Africa

DOI:

https://doi.org/10.13052/jgeu0975-1416.1321

Keywords:

Travelling salesman tour, shortest connected graph, index restricted shortest connected graph, travelling salesman heuristic, nearest-neighbour heuristic to the travelling salesman

Abstract

Nearest neighbour approach for determination of the minimum travelling salesman tour is a well-documented heuristic for a quick identification of the salesman tour and its associated cost. This heuristic has been used on large sized networks for a quick approximate solution to the travelling salesman problem. For the last several years, an alternative approach to the minimum travelling salesman tour has been developed through the minimum spanning tree (MST). The MST was converted to an index restricted MST (IRMST) and that IRMST was used to find the minimum travelling salesman tour. These two approaches, i.e., the greedy heuristics and the index restricted minimum spanning tree seems to have stronger relations and it is this relationship, which has been explored in this paper. The nearest neighbour approach to find the travelling salesman tour has been modified by incorporating the index balancing theorem. A further modification to this heuristic has been suggested to identify the minimum travelling salesman tour. Many small size problems have been attempted using the heuristic discussed in this paper, and it resulted in an optimal solution but an analytical proof for optimality remains a challenge. This approach further strengthens a natural question about the ‘NP Hard’ category associated with the minimum travelling salesman tour problem.

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Author Biographies

Santosh Kumar OAM, Department of Mathematical and Geospatial Sciences, School of Sciences, RMIT University, Melbourne Australia

Santosh Kumar OAM received PhD degree in Operations Research from Delhi University. He is author and co-author of over 220 papers and 4 books in Operations Research. His contributions in the field of OR were recognized by the Australian Society for Operations Research (ASOR)in 2009 by their ‘Ren Pots’ award. He has received a recognition award in 2011 for contributions in OR in Southern African region from the South African OR Society as a non-member of the society and a non-resident of South Africa. He was the President of the Asia Pacific Operations Research societies (1995–97), where ASOR was a member along with 7 other countries in the region. He is currently an Honorary Professor at RMIT University, Melbourne. He is a Chartered Mathematician and a Fellow of the Institute of Mathematics and its Applications, UK. On 14th June 2021, he was awarded under the Australian award system a medal of the Order of Australia (OAM).

Elias Munapo, School of Economics and Decision Sciences, North West University, Mafikeng 2735, South Africa

Elias Munapo holds a BSc. (Hons) Applied Mathematics (1997), MSc. Operations Research (2002) and a PhD Operations Research (2010), all from the National University of science and Technology (N.U.S.T.), Zimbabwe. A certificate in outcomes-based assessment in Higher Education and Open distance learning, University of South Africa (UNISA), certificate in University Education Induction Program, University of KwaZulu-Natal (UKZN). He is a Professional Natural Scientist certified by the South African Council for Natural Scientific Professions (SACNASP), 2012; and NRF rated in South Africa. He has published/co-published over 140 articles and two books. He has also edited/co-edited 7 books and is a guest editor of Applied Sciences, Algorithms and Next Energy journals which are under MDPI. He has supervised/co-supervised eleven doctoral students and over 30 students at Master level. He is a member of the Operations Research Society of South Africa, Executive Committee Member 2012–13, South African Council for Natural Scientific Professions (SACNASP) as a Certified Natural Scientist, European Conference on Operational Research (EURO) and the International Federation of Operations Research Societies (IFORS), and a member of the International Conference on Optimization (ICO) and is part of the team preparing to host ICO 2026 at Johannesburg.

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Published

2025-06-02

How to Cite

OAM, S. K., & Munapo, E. (2025). A Greedy Reconstruction Heuristic for Solving the Minimum Travelling Salesman Tour Problem. Journal of Graphic Era University, 13(02), 221–244. https://doi.org/10.13052/jgeu0975-1416.1321

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