OPSEARCH

, Volume 55, Issue 1, pp 150–164

# A minimum spanning tree based heuristic for the travelling salesman tour

• Santosh Kumar
• Elias Munapo
• ‘Maseka Lesaoana
• Philimon Nyamugure
Theoretical Article

## Abstract

This paper presents a heuristic to find the travelling salesman tour (TST) in a connected network. The approach first identifies a node and two associated arcs that are desirable for inclusion in the required TST. If we let this node be denoted by $$p$$ and two selected arcs emanating from this node be denoted by $$\left( {p,q} \right)\,{\text{and}}\,\left( {p,k} \right),$$ then we find a path joining the two nodes $$q\,{\text{and}}\, k$$ passing through all the remaining nodes of the given network. A sum of these lengths, i.e. length of the links $$\left( {p,q} \right)\,{\text{and}}\,\left( {p,k} \right)$$ along with the length of the path that joins the nodes $$q\,{\text{and}}\, k$$ passing through all the remaining nodes will result in a feasible TST, hence gives an upper bound on the TST. A simple procedure is outlined to identify: (1) the node $$p$$, (2) the two corresponding links $$\left( {p,q} \right)\,{\text{and}}\,\left( {p,k} \right),$$ and (3) the path joining the nodes $$q\,{\text{and}}\, k$$ passing through all the remaining nodes. The approach is based on the minimum spanning tree; hence the complexity of the TST is reduced. The network in the present context has been assumed to be a connected network with at least two arcs emanating from each node.

## Keywords

Connected network Minimum spanning tree path Travelling salesman tour

## Notes

### Acknowledgements

Authors are grateful to the referees for their constructive suggestions and helpful comments.

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© Operational Research Society of India 2017

## Authors and Affiliations

• Santosh Kumar
• 1
• Elias Munapo
• 2
• ‘Maseka Lesaoana
• 3
• Philimon Nyamugure
• 4
1. 1.Department of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
2. 2.School Economics and Decision SciencesNorth West University, Mafikeng CampusMafikengSouth Africa
3. 3.Department of Statistics and Operations ResearchUniversity of LimpopoSovengaSouth Africa
4. 4.Department of Statistics and Operations ResearchNational University of Science and TechnologyAscot, BulawayoZimbabwe