Using Gavish-Grave LP to Formulate the Directed Black and White Traveling Salesman Problem

The black and white traveling salesman problem (BWTSP) is a new class of NP-hard problem arising from work on airline scheduling and telecommunication fiber networks. The existing Ghiani LP for the undirected BWTSP contains an exponential number of constraints. For a special case of the directed BWTSP whose L = +∞ , the LP with polynomial number of constraints could be obtained by transforming it to an asymmetric traveling salesman problem with replenishment arcs (RATSP), whereas there exists no LP for the directed BWTSP in its general form. This paper proposes a LP with 2 3 2 n n + constraints only for the directed BWTSP in such a way that, by reducing the problem to an asymmetric traveling salesman problem (ATSP), we add 2 n cardinality constraints and 2 n length constraints to the existing GavishGrave LP for the ATSP. The new LP is also valid for the undirected BWTSP when viewed as a special case of the directed BWTSP.