  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 $\triangle$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.
