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The World TSP: a tour through 1,904,711 cities
Shorten the best tour through 1,904,711 populated places on Earth, now within 0.0471% of optimal, or raise the lower bound set in 2007.
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| 1 | +# The World TSP: a tour through 1,904,711 cities | |
| 2 | + | |
| 3 | +The World TSP asks for the shortest round trip through 1,904,711 populated places on Earth. The best tour, found in October 2025, is known to be at most 0.0471% longer than an optimal one; the lower bound behind that guarantee has not moved since 2007. | |
| 4 | + | |
| 5 | +## Problem | |
| 6 | + | |
| 7 | +The instance lists 1,904,711 locations by latitude and longitude. The distance between two of them is an approximation of the great-circle distance on a spherical Earth, given in metres; the Waterloo pages call it the GEOM norm, a variant of the TSPLIB `GEO` norm adapted to decimal degrees, and publish a fragment of C code that computes it. Find a tour, a cyclic order visiting every location exactly once, of minimum total length, or prove that a given tour is optimal. | |
| 8 | + | |
| 9 | +## Background | |
| 10 | + | |
| 11 | +The instance was assembled on the Waterloo TSP pages as a large-scale challenge from two gazetteers: the National Imagery and Mapping Agency database of geographic feature names and the Geographic Names Information System. It contains every location registered as a populated place, plus several research bases in Antarctica. The National TSP instances are taken from the same data. Lower bounds come from Concorde, the linear-programming-based solver described by Applegate, Bixby, Chvátal and Cook (2006); the record tours come from heuristics, chiefly Helsgaun's LKH and Nagata's genetic algorithm with edge assembly crossover (GA-EAX). | |
| 12 | + | |
| 13 | +## What is known | |
| 14 | + | |
| 15 | +- Best tour: 7,515,755,912, found by Yuichi Nagata on 17 October 2025 by adding the previous record to the population of his GA-EAX heuristic. It improved Keld Helsgaun's tour of 7,515,755,956 (15 February 2021) by 44 metres. | |
| 16 | +- Best lower bound: 7,512,218,268, established by Concorde on 5 June 2007 with CPLEX as the linear-programming solver. The bound shows that the record tour is at most 0.0471% longer than an optimal one. | |
| 17 | +- Progress has slowed: from October 2011 to June 2020 the record improved by only 7,604 metres, while two tours in February 2021 lowered it by 14,628 metres. Earlier records go back to a 2003 tour of 7,517,285,610. | |
| 18 | +- The page (last updated 24 October 2025) invites improved tours and improved lower bounds, and offers the three best tours for download. | |
| 19 | + | |
| 20 | +## What would count as progress | |
| 21 | + | |
| 22 | +- A tour in TSPLIB tour format shorter than 7,515,755,912. A script checks that it is a permutation of the 1,904,711 cities and recomputes its length with the published GEOM distance code in seconds. | |
| 23 | +- A lower bound above 7,512,218,268, backed by a checkable certificate (for example the cutting planes and branching that produce it, as in the certified solution of `pla85900`, Applegate et al., 2009). Proving any tour optimal is a separate and far harder kind of progress. | |
| 24 | + | |
| 25 | +## References | |
| 26 | + | |
| 27 | +1. D. L. Applegate, R. E. Bixby, V. Chvátal and W. J. Cook (2006). *The Traveling Salesman Problem: A Computational Study*. Princeton University Press. https://press.princeton.edu/books/hardcover/9780691129938/the-traveling-salesman-problem | |
| 28 | +2. Y. Nagata and S. Kobayashi (2013). A powerful genetic algorithm using edge assembly crossover for the traveling salesman problem. *INFORMS Journal on Computing* 25, 346–363. https://doi.org/10.1287/ijoc.1120.0506 | |
| 29 | +3. K. Helsgaun (2000). An effective implementation of the Lin–Kernighan traveling salesman heuristic. *European Journal of Operational Research* 126, 106–130. https://doi.org/10.1016/S0377-2217(99)00284-2 | |
| 30 | +4. D. L. Applegate, R. E. Bixby, V. Chvátal, W. Cook, D. G. Espinoza, M. Goycoolea and K. Helsgaun (2009). Certification of an optimal TSP tour through 85,900 cities. *Operations Research Letters* 37, 11–15. https://doi.org/10.1016/j.orl.2008.09.006 | |
| 31 | + | |
| 32 | +## Source | |
| 33 | + | |
| 34 | +Imported from [Waterloo TSP pages, World TSP](https://www.math.uwaterloo.ca/tsp/world/index.html), curated by William Cook. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion. | |