Approximation Algorithms for the Directed k-Tour and k-Stroll Problems

We consider two natural generalizations of the Asymmetric Traveling Salesman problem: the k-Stroll and the k-Tour problems. The input to the k-Stroll problem is a directed n-vertex graph with nonnegative edge lengths, an integer k, and two special vertices s and t. The goal is to find a minimum-length s-t, walk, containing at least k distinct vertices. The k-Tour problem can be viewed as a special case of k-Stroll, where S = t. That is, the walk is required to be a tour, containing some pre-specified vertex s. When k = n, the k-Stroll problem becomes equivalent to Asymmetric Traveling Salesman Path, and k-Tour to Asymmetric Traveling Salesman. Our main result is a polylogarithmic approximation algorithm for the k-Stroll problem. Prior to our work, only bicriteria (O(log(2) k), 3)-approximation algorithms have been known, producing walks whose length is bounded by 3OPT, while the number of vertices visited is Omega(k/log(2) k). We also show a simple O(log(2) n/log log n)-approximation algorithm for the k-Tour problem. The best previously known approximation algorithms achieved min (O(log(3) k), O(log(2) n.log k/log log n))-approximation in polynomial time, and O(log(2) k)-approximation in quasipolynomial time.