A Method to Compute the Sparse Graphs for Traveling Salesman Problem Based on Frequency Quadrilaterals

Abstract: In this paper, an iterative algorithm is designed to compute the sparse graphs for traveling salesman problem (TSP) according to the frequency quadrilaterals so that the computation time of the algorithms for TSP will be lowered. At each computation cycle, the algorithm first computes the average frequency \bar{f}(e) of an edge e with N frequency quadrilaterals containing e in the input graph G(V,E). Then the 1/3|E| edges with low frequency are eliminated to generate the output graph with a smaller number of edges. The algorithm can be iterated several times and the original optimal Hamiltonian cycle is preserved with a high probability. The experiments demonstrate the algorithm computes the sparse graphs with the O(nlog_2n) edges containing the original optimal Hamiltonian cycle for most of the TSP instances in the TSPLIB. The computation time of the iterative algorithm is O(Nn^2).