Valid inequalities for mixed-integer programmes with fixed charges on sets of variables

We consider mixed 0-1 linear programs in which one is given a collection of (not necessarily disjoint) sets of variables and, for each set, a fixxed charge is incurred if and only if at least one of the variables in the set takes a positive value. We derive strong valid linear inequalities for these problems, and show that they generalise and dominate a subclass of the well-known flow cover inequalities for the classical fixed-charge problem