Lifted cover inequalities are well-known cutting planes for 0-1 linear programs. We show how one of the earliest lifting procedures, due to Balas, can be significantly improved. The resulting procedure has some unusual properties. For example, (i) it can yield facet-defining inequalities even if the given cover is not minimal, (ii) it can yield facet-defining inequalities that cannot be obtained by standard lifting procedures, and (iii) the associated superadditive lifting function is integer-valued almost everywhere.
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 fixed 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.
The most effective software packages for solving mixed 0-1 linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions.
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