Improved Approximation Guarantees for Minimum-Weight k-Trees And Prize-Collecting Salesmen

Consider a salesperson that must sell some quota of brushes in order to win a trip to Hawaii. This salesperson has a map (a weighted graph) in which each city has an attached demand specifying the number of brushes that can be sold in that city. What is the best route to take to sell the quota while traveling the least distance possible? Notice that unlike the standard traveling salesman problem, not only do we need to figure out the order in which to visit the cities, but we must decide the more fundamental question: which cities do we want to visit? In this paper we give the first approximation algorithms with poly-logarithmic performance guarantees for this problem, as well as for the slightly more general PCTSP problem of Balas, and a variation we call the "bank-robber problem" (also called the "orienteering problem" by Golden, Levi, and Vohra). We do this by providing an O(log 2 k) approximation to the k-MST problem which is defined as follows. Given an undirected graph on n nod..