 
Summary: Improved Approximation Guarantees for
MinimumWeight kTrees and PrizeCollecting
Salesmen
Baruch Awerbuch \Lambda Yossi Azar y Avrim Blum z
Santosh Vempala z
Abstract
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, and approximate as well the slightly more
general ``Prize Collecting Traveling Salesman Problem'' (PCTSP) of Balas, and a vari
ation we call the ``Bankrobber Problem'' (also called the ``Orienteering Problem'' by
Golden, Levi, and Vohra). We do this by providing an O(log 2 k) approximation to the
somewhat cleaner kMST problem which is defined as follows. Given an undirected
