Multiple Traveling Salesmen in Asymmetric Metrics

Abstract

We consider some generalizations of the Asymmetric Traveling Salesman Path Problem. In these variants, we have multiple salesmen that we are to move around a metric and the goal is to have each node visited by at least one salesman. This should be done while minimizing the total distance travelled by all salesmen.

In the first variant, we are given two nodes s,t and an integer k and the goal is to find k paths from s to t whose union covers all nodes. We give an efficient bicriteria approximation for this problem that uses at most k + k/b paths of total length at most O(b log|V|) times the optimum value of a natural LP relaxation. By setting b appropriately, we can obtain a true O(k logn)-approximation, an O(logn)-bicriteria approximation using at most 2k paths, or, more generally, an $O(\frac{1}{\epsilon} \log n)$-bicriteria approximation using at most (1 + ε)k paths. Prior to this work, only an O(k ² logn)-approximation and an O(logn)-bicriteria approximation using at most O(k logn) paths were known.

Next, we consider the case where we have k pairs of nodes $\{(s_i, t_i)\}_{i=1}^k$. 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(logn) when k = 2. On the other hand, we demonstrate that the general problem cannot be approximated within any finite ratio unless P = NP.