Approximation Hardness of TSP with Bounded Metrics

The general asymmetric TSP with triangle inequality is known to be approximable only within an O(log n) factor, and is also known to be approximable within a constant factor as soon as the metric is bounded. In this paper we study the asymmetric and symmetric TSP problems with bounded metrics, i.e., metrics where the distances are integers between one and some upper bound B. We first prove approximation lower bounds of 321/320 and 741/740 for the asymmetric and symmetric TSP with distances one and two, improving over the previous best lower bounds of 2805/2804 and 5381/5380. Then we consider the TSP with triangle inequality and distances that are integers between one and eight and prove approximation lower bounds of 131/130 for the asymmetric and 405/404 for the symmetric, respectively, version of that problem, improving over the previous best lower bounds of 2805/2804 and 3813/3812 by an order of magnitude.