The Traveling Salesman Problem (TSP) is traditionally studied through the lens of algorithmic performance and optimal tour certification. Once an instance is solved optimally, it is commonly regarded as exhausted from a scientific standpoint. In this work, we revisit the classical TSPLIB instance xqf131, whose optimal tour length has been known since its exact resolution by Concorde in the early 2000s. Rather than proposing a new solver or revisiting optimality proofs, we focus on a different object of study: the internal structure of the set of optimal tours itself. By analyzing a large corpus of independently generated optimal tours, we show that optimality for xqf131 is neither unique nor diffuse. Instead, the optimal solutions form a finite and highly structured family, characterized by a rigid backbone of invariant edges and a small number of localized permutation modules. We introduce the notion of the Meghezzien, defined as the superposition of all observed optimal tours, as a compact representation of this structure. This study provides an empirical characterization of the optimal solution space of a classical TSP instance, highlighting the relevance of structural analysis beyond solver-centric perspectives. While the results are instance-specific and do not claim generality, they suggest that optimality in combinatorial optimization problems may be more naturally understood as a structured object rather than a single solution.
Axel Abdel Rahamane MEGHEZZI (Sun,) studied this question.