Minimum directed 1-subtree relaxation for score orienteering problem

Score orienteering is a variant of the orienteering game where competitors start at a specified control point, visit as many control points as possible and thereby collect prizes, and return to the starting point within a prescribed amount of time. The competitor with the highest amount of prizes wins the game. In this paper, we propose the minimum directed 1-subtree problem as a new relaxation to this problem and develop two methods to improve its lower bound. We describe a cut and dual simplex method and a Lagrangian relaxation method, and construct an algorithm that combines these two methods in an appropriate way. Computational experiments are carried out to analyse the behavior of the proposed relaxation with respect to the characteristics of the test problems. Especially for large-scale instances, we find that the proposed relaxation is superior to the ordinary assignment relaxation.