Approximations for ATSP with Parametrized Triangle Inequality

We give a constant factor ( ?/1-? + ?) approximation for the asymmetric traveling salesman problem in graphs with costs on the edges satisfying ?-parametrized triangle inequality (?-Asymmetric graphs) for ? ? [1/2, 1). We also give an improvement of the algorithm with approximation factor approaching ?/1-?.We also explore the cmax/cmin ratio of edge costs in a general asymmetric graph. We show that for ? ? [1/2, 1/?3 ), cmax/cmin ? 2?3/1-3?2, while for ? ? [ 1/?3, 1), this ratio can be arbitrarily large. We make use of this result to give a better analysis to our main algorithm. We also observe that when cmax/cmin > ?2/1-?-?2 with ? ? (1/2, ?5-1/2), the minimum cost and the maximum cost edges in the graph are unique and are reverse to each other.