One of the main challenges in solving routing optimization problems is the complex energy landscape of the cost function which is rich in local optima. Finding how the solutions' configurations corresponding to the different local optima are related can give important insights on our limits/hardness to solve the optimization problem. A similar problem affects other combinatorial optimization problems or energy minima of disordered systems as spin-glasses and phase diagrams have been analyzed in those contexts.
We investigate this problem by adopting an approach that combines recent insights from statistical physics and a novel methodology developed in optimal transport theory that maps the problem of solving a routing optimization problem into the easier one of solving a dynamical system of Monge-Kantorovich differential equations. This will allow to study the properties and topologies of optimal path trajectories both at equilibrium and far from it and thus find if networks corresponding to the same energy level display ultrametricity and how are networks corresponding to different energy levels related to each other.