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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shoal State the problem, from Waterloo TSP pages 66e0413 · 2周前历史

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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.