Abstract. Store-and-forward packet routing belongs to the most fundamen-tal tasks in network optimization. Limited bandwith requires that some pack-ets cannot move to their destination directly but need to wait in intermediate nodes on their path or take detours. It is desirable to find schedules that en-sure fast delivery of the packets, in particular for time critial applications. In this paper we investigate the packet routing problem theoretically. We prove that the problem cannot be approximated with a factor of 6/5 − for all > 0 unless P = NP. We show that restricting the graph topology to planar graphs or even directed trees still does not allow a PTAS. We present three individual algorithms for packet routing on trees: For undirected trees we give a factor 2 approximation. For directed trees we show that the path coloring problem can be solved optimally in polynomial time. Based on this, we show how to construct a schedule with length C + D − 1 where C and D denote the trivial lower bounds congestion and dilation. If the lengths of the paths of the packets are pairwise different, we can compute a schedule of length D on directed trees which is optimal. For all cases we show that our analysis is tight. Finally, we show that it is NP-hard to approximate the packet routing problem with an absolute error of k for any k ≥ 0. 1
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