Date: 01 Jun 2005

0/1-Integer programming: Optimization and Augmentation are equivalent

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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 x new\(\mathcal{F}\)with cx new > cx 0 (Augmentation), and

  • find a feasible point xnew\(\mathcal{F}\)with cx new > c x 0 such that x newx 0 is “irreducible” (Irreducible Augmentation).

This generalizes results and techniques that are well known for 0/1-integer programming problems that arise from various classes of combinatorial optimization problems.