We investigate algorithms that solve exactly the robust single machine scheduling problem that minimizes the total tardiness. We model the processing times as uncertain and let them take any value in a budgeted uncertainty set. Therefore , the objective seeks to minimize the worst-case tardiness over all possible values. We compare, through computational experiments, two types of solution algorithms. The first combines classical MILP formulations with row-and-column generation algorithms. The second generalizes the classical branch-and-bound algorithms to the robust context, extending state-of-the-art concepts used for the deterministic version of the problem. By generalizing the classical branch-and-bound algorithm we are able to assemble and discuss good algorithmic decisions steps that once put together make our robust branch-and-bound case attractive. For example, we extend and adapt dominance rules to our uncertain problem, making them an important component of our robust algorithms. We assess our algorithms on instances inspired by the scientific literature and identify under what conditions an algorithm has better performance than others. We introduce a new classifying parameter to group our instances, also extending existing methods for the deterministic problem case.