0/1-Integer Programming: Optimization and Augmentation are Equivalent

For every family of sets \(\mathcal{F} \subseteq \{ 0,1\} ^n\)the following problems are strongly polynomial time equivalent: given a feasible point x0 ∈ \(\mathcal{F}\)and a linear objective function c ∈ ℤ n , find a feasible point x*∈\(\mathcal{F}\)that maximizes c x (Optimization), find a feasible point xnew ∈ \(\mathcal{F}\)with cxnew > cx0 (Augmentation), and find a feasible point xnew ∈ \(\mathcal{F}\)with cxnew > c x0 such that xnew−x0 is “irreducible” (Irreducible Augmentation).