Prize-Collecting Traveling Salesman and Related Problems

Abstract

The most general version of the Prize Collecting Traveling Salesman Problem (PCTSP) was first introduced by Balas [8]. In this problem, a salesman has to collect a certain amount of prizes (the quota) by visiting cities. A known prize can be collected in every city. Furthermore, by not visiting a city, the salesman incurs a pecuniary penalty. The goal is to minimize the total travel distance plus the total penalty, while starting from a given city and collecting the quota. The problem generalizes both the Quota TSP, which is obtained when all the penalties are set to zero, and the Penalty TSP (sometimes unfortunately also called PCTSP), in which there is no required quota, only penalties. A special case of the Quota TSP is the k-TSP, in which all prizes are unitary (k is the quota). The k-TSP is strongly tied to the problem of finding a tree of minimum cost spanning any k vertices in a graph, called the k-MST problem. The k-MST and the k-TSP are NP-hard. They have been the subject of several studies for good approximation