An iterative algorithm to eliminate edges for traveling salesman problem based on a new binomial distribution

Traveling salesman problem (TSP) is one of the extensively studied NP-hard problems. The recent research showed that the TSP on sparse graphs could be resolved in the relatively shorter computation time than that on the complete graph Kn$K_{n}$. This paper updates a previous probability model for the optimal Hamiltonian cycle edges according to the frequency quadrilaterals in Kn$K_{n}$. A new binomial distribution for TSP is rebuilt to show the probability that an edge e has the frequency 5 in a frequency quadrilateral. Based on the binomial distribution, an iterative algorithm is designed to compute the sparse graphs for TSP. There are two steps at each computation cycle. Firstly, N frequency quadrilaterals containing an edge e in the input graph is chosen to compute the average frequency f̄(e)$\bar {f}(e)$ with the frequency quadrilaterals where e has the frequency 5. Secondly, half edges with the small values f̄(e)$\bar {f}(e)$ are eliminated. The two steps are repeated until a sparse graph is computed. The computation time of the algorithm is O(Nn2)$O(Nn^{2})$. For the TSP instances in the TSPLIB, the experimental results illustrated that the sparse graphs with the O(nlog2n)$O(n\log _{2} n)$ edges are computed and the original optimal solution is preserved. The experiments means the optimal Hamiltonian cycle edges have the bigger average frequency f̄(e)$\bar {f}(e)$ in Kn$K_{n}$ and the subgraphs of Kn$K_{n}$ so they are preserved in the computation process.