Title: Introduction of an arc rank order statistic for analysis of optimal solutions in travelling salesman problems

Authors: Jerry L. Shaw; Donovan Fuqua; Hansuk Sohn; Manuel Ivan Rodriguez Borbon

Addresses: New Mexico State University, 2850 Weddell Street Las Cruces, NM, 88003, USA ' New Mexico State University, 2850 Weddell Street Las Cruces, NM, 88003, USA ' New Mexico State University, 2850 Weddell Street Las Cruces, NM, 88003, USA ' New Mexico State University, 2850 Weddell Street Las Cruces, NM, 88003, USA

Abstract: This research provides an empirically derived and previously unreported insight into the actual composition and common features of optimal solutions in travelling salesman problems (TSPs). One unreported characteristic of these solutions is the frequency with which a vertex is connected to another relatively near or distant vertex. Based on a mathematical question on optimal solution arc lengths, we introduce an arc rank statistic and use it to analyse optimal solutions for a range of standard TSPs. The resulting analysis shows, with few exceptions, that the frequency with which low-ranked arcs appear is a decreasing function. Further, the frequency distributions of arc ranks generally have fat tails. Log/log transforms of the frequency distributions were analysed to show linear relations; suggesting that the distribution of arc ranks in optimal TSP solutions generally follows a power law distribution. This knowledge suggests that statistical information about a solution may be sufficient to inform an observer about the quality of a solution.

Keywords: optimisation; travelling salesman problem; TSP; graph theory; edge ranking.

DOI: 10.1504/IJMOR.2025.145754

International Journal of Mathematics in Operational Research, 2025 Vol.30 No.3, pp.308 - 328

Received: 25 Jul 2023
Accepted: 27 Jul 2023
Published online: 23 Apr 2025 *

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