We consider some generalizations of the Asymmetric Traveling Salesman Path problem. Suppose we have an asymmetric metric G = (V,A) with two distinguished nodes s,t. We are also given a positive integer k. The goal is to find k paths of minimum total cost from s to t whose union spans all nodes. We call this the k-Person Asymmetric Traveling Salesmen Path problem (k-ATSPP). Our main result for k-ATSPP is a bicriteria approximation that, for some parameter b >= 1 we may choose, finds between k and k + k/b paths of total length O(b log |V|) times the optimum value of an LP relaxation based on the Held-Karp relaxation for the Traveling Salesman problem. On one extreme this is an O(log |V|)-approximation that uses up to 2k paths and on the other it is an O(k log |V|)-approximation that uses exactly k paths. Next, we consider the case where we have k pairs of nodes (s_1,t_1), ..., (s_k,t_k). The goal is to find an s_i-t_i path for every pair such that each node of G lies on at least one of these paths. Simple approximation algorithms are presented for the special cases where the metric is symmetric or where s_i = t_i for each i. We also show that the problem can be approximated within a factor O(log n) when k=2. On the other hand, we demonstrate that the general problem cannot be approximated within any bounded ratio unless P = NP.

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