Counter-Example to Diaby’s et al. Linear Programming Solution to the Traveling Salesman Problem

This article presents a method of constructing counter-examples and a complete counter-example to the linear programming model alleged to be the solution to the traveling salesman problem. The counter-example is checked against the model proposed by Diaby et al. However, it applies to all similar formulations of the TSP problem.

Although the model in question was published in 2006, and there were several discussions regarding its correctness, the counter-example was never presented.

The presented counter-example is a regular graph, and the aim was not to have an example with the least possible size; therefore, the focus was on clarity. The counter-example has, therefore, 366 nodes in two main clusters, each node (in the main part) having exactly four connections to other nodes in the cluster.