Approximating the asymmetric profitable tour

We study the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. In [3], the authors defined it as the Profitable Tour Problem (PTP). We present an (1 + log(n))-approximation algorithm for the asymmetric PTP with n is the vertex number. The algorithm that is based on Frieze et al.’s heuristic for the asymmetric traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers (feasible solution for the original problem), represents a directed version of the Bienstock et al.’s [2] algorithm for the symmetric PTP.