Maximal Closure of a Graph and Applications to Combinatorial Problems

This paper generalizes the selection problem discussed by J. M. Rhys [Rhys, J. M. W. 1970. Shared fixed cost and network flows. Management Sci. 17 (3, November).], J. D. Murchland [Murchland, J. D. 1968. Rhys's combinatorial station selection problem. London Graduate School of Business Studies, Transport Network Theory Unit, Report LBS-TNT-68, June 10.], M. L. Balinski [Balinski, M. L. 1970. On a selection problem. Management Sci. 17 (3, November).] and P. Hansen [Hansen, P. 1974. Quelques approches de la programmation non lineaire en variables 0-1. Conference on Mathematical Programming, Bruxelles, May.]. Given a directed graph G, a closure of G is defined as a subset of nodes such that if a node belongs to the closure all its successors also belong to the set. If a real number is associated to each node of G a maximal closure is defined as a closure of maximal value.