On Prize-CollectingTours and the Asymmetric Travelling Salesman Problem

We consider a special version of the Travelling Salesman Problemwhich is to determine a tour visitingeach vertex in the graph at most one time;if a vertex is left unrouted a given penalty has to be paid.The objectivefunction is to find a balance between these penalties and the cost of thetour. We call this problem the Profitable Tour Problem (PTP). If,in addition, to each vertex is associated a price andthere is a knapsack constraint which guarantees that a sufficiently largeprice is collected, we have the well knownPrice-Collecting Travelling Salesman Problem (PCTSP).In this paper we summarize the main results presented in the literature,then we give lower bounds for the asymmetric version of PTP and PCTSP. Moreoverwe show, through computational experiments, that large size instances of theAsymmetric PTP can be solved exactly.