In this paper we introduce three greedy algorithms for the traveling salesman problem. These algorithms are unique in that they use arc tolerances, rather than arc weights, to decide whether or not to include an arc in a solution. We report extensive computational experiments on benchmark instances that clearly demonstrate that our tolerance-based algorithms outperform their weight-based counterpart. Along with other papers dealing with the Assignment Problem, this paper indicates that the potential for using tolerance-based algorithms for various optimization problems is high and motivates further investigation of the approach. We recommend one of our algorithms as a significantly better alternative to the weight-based greedy, which is often used to produce initial TSP tours.