A Representation Model for TSP

Abstract

Traveling salesman problem (TSP) has been proven to be NP-complete and it is regarded for more than half a century. It is often represented as a weighted graph whereas the weighted graph cannot provide enough heuristic information for TSP. We do not know which edges belong to the best solution according to the edges' weights. Here the frequency graph is introduced as a novel representation model for TSP. The frequency graph includes the law to connect the edges into the optimal Hamiltonian cycle (OHC). Based on the frequency graph, a sparse graph is computed and the search space of the OHC is reduced. Three steps are necessary to convert a complete weighted graph into a sparse graph with small number of edges. The frequency graph is computed with a set of local optimal Hamiltonian paths (LOHPs) at first. A sparse graph is computed according to a defined frequency threshold at the second step. A sparser graph is computed with the former sparse graph at the third step. The average degree of the final sparse graph is bounded to a small number and the search space of TSP is reduced. At last, the representation model is verified with several TSP instances with known optimal solutions.