New approximation guarantees for minimum-weight κ-trees and prize-collecting salesmen

We consider a formalization of the following problem. A salesperson 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 algorithm having a polylogarithmic performance guarantee for this problem, as well as for the slightly more general "prize-collecting traveling salesman problem" (PCTSP) 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(log² κ) approximation to the somewhat cleaner κ-MST problem which is defined as follows. Given an undirected graph on n nodes with nonnegative edge weights and an integer κ ≤ n, find the tree of least weight that spans κ vertices. (If desired, one may specify in the problem a "root vertex" that must be in the tree as well.) Our result improves on the previous best bound of O(√κ) of Ravi et al.