We consider online routing optimization problems where the objective is to minimize the time needed to visit a set of locations under various constraints; the problems are online because the set of locations are revealed incrementally over time. We make no probabilistic assumptions whatsoever about the problem data. We consider two main problems: (1) the online Traveling Salesman Problem (TSP) with precedence and capacity constraints and (2) the online TSP with m salesmen. For both problems we propose online algorithms, each with a competitive ratio of 2; for the m-salesmen problem, we show our result is best-possible. We also consider polynomial-time online algorithms as well as various generalizations of our results.
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