Obtaining Quality-Proved Near Optimal Results for Traveling Salesman Problem

The traveling salesman problem (TSP) is one of the most challenging NP-hard problems. It has widely applications in various disciplines such as physics, biology, computer science and so forth. The best known approximation algorithm for Symmetric TSP (STSP) whose cost matrix satisfies the triangle inequality (called $\mathrm{△}$STSP) is Christofides algorithm which was proposed in 1976 and is a $\frac{3}{2}$-approximation. Since then no proved improvement is made and improving upon this bound is a fundamental open question in combinatorial optimization. In this paper, for the first time, we propose Truncated Generalized Beta distribution (TGB) for the probability distribution of optimal tour lengths in a TSP. We then introduce an iterative TGB approach to obtain quality-proved near optimal approximation, i.e., (1+$\frac{1}{2}(\frac{\alpha +1}{\alpha +2}{)}^{K-1}$)-approximation where $K$ is the number of iterations in TGB and $\alpha (>>1)$ is the shape parameters of TGB. The result can approach the true optimum as $K$ increases.