Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs

In metric asymmetric traveling salesperson problems the input is a complete directed graph in which edge weights satisfy the triangle inequality, and one is required to ¯nd a minimum weight walk that visits all vertices. In the asymmetric traveling salesperson problem (ATSP) the walk is required to be cyclic. In asymmetric traveling salesperson path problem (ATSPP), the walk is required to start at vertex s and to end at vertex t. We improve the approximation ratio for ATSP from 4 3 log3 n ‘ 0:84 log2 n to 2 3 log2 n. This improvement is based on a modification of the algorithm of Kaplan et al [JACM 05] that achieved the previous best approximation ratio. We also show a reduction from ATSPP to ATSP that loses a factor of at most 2 + ² in the approximation ratio, where ² > 0 can be chosen to be arbitrarily small, and the running time of the reduction is polynomial for every fixed ². Combined with our improved approximation ratio for ATSP, this establishes an approximation ratio of ( 4 3 + ²) log2 n for ATSPP, improving over the previous best ratio of 4 loge n ‘ 2:76 log2 n of Chekuri and Pal [Approx 2006].