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 [DellʹAmico, M., Maffioli, F. and Väbrand P.: On Prize-collecting Tours and the Asymmetric Travelling Salesman Problem, Int. Trans. Opl Res., Vol. 2, 297–308 (1995)], 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.ʹs 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.ʹs [Bienstock, D., Goemans, M.X., Simchi-Levi, D. and Williamson, D.P.: A note on the prize collecting traveling salesman problem, Math. Prog., Vol. 59, 413–420 (1993)] algorithm for the symmetric PTP.