Multiple Traveling Salesmen in Asymmetric Metrics

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 ∈ V . 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 {(si, ti)}i=1. The goal is to find an si− ti 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 si = ti 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. ∗Research supported by an iCORE ICT/AITF award while studying at the University of Alberta.