A Travelling Salesman Problem (TSP) is a well known computational challenge. A lot of algorithms were developed to solve it or its special cases.

I came around an article authored by Fang Liu 'A dual population parallel ant colony optimization algorithm for solving the travelling salesman problem'. In this article he proposed a modification of an Ant Colony System algorithm for solving TSP and presented results obtained by his algorithm. In the table with results all looked fine - his algorithm was able to provide very good solutions for the TSP instances from TSPLIB (which is the common testing ground for TSP algorithms).

So the researcher presented good results... it seems. But then he decided to show

**his algorithm was able to find and annotated them with the corresponding route costs. Lets take a look at one of them. Here you are his**__the best routes__**for the 'att48' instance from the TSPLIB:**__best route__Route that claims to be optimal (but the cost is very wrong) |

The optimal route for 'att48' and its cost is well-known (it applies to all TSPLIB instanses). Its cost is approximately 33523 (there are different approaches to rounding distances between points). So what we see at the picture above should be the optimal route (or extremely close to it). But dear reader, do you think that you see optimal route? Humans are able to provide very good solutions to TSP instances that consists of not too many points. I bet you can draw far better route yourself. The route from this picture is 1, 8, 46, 33, 20, 17, 43, 27, 19, 37, 6, 30, 36, 28, 7, 18, 44, 31, 38, 9, 40, 15, 12, 11, 47, 21, 13, 25, 14, 23, 3, 22, 16, 41, 34, 2, 29, 5, 48, 39, 32, 24, 42, 10, 45, 35, 4, 26, 1 and its cost is 41052 which is whooping 22% far from optimal! The same story for another illustration in the article.

Here take a look at the optimal route which cost is really 33523:

Actually optimal route for 'att48' with Cost = 33523 |

So what we can conclude? I do believe that the routes demonstrated in the article are the best routes found by given algorithm, but costs are put-up for the routes and for the table of results in the article as well. I think that author developed an algorithm that wasn't able to find good solutions and provided fraud table with the put-up testing results. And clearly this article wasn't reviewed by scientist that have knowledge in TSP area because these plots are so obviously flawed that one can't overlook it!

No one usually shows plots of the routes they find with their algorithms. I wonder if there are more modern algorithms with the put-up results?