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.

    Andreas S. Schulz has been supported by the graduate school “Algorithmische Diskrete Mathematik”. The graduate school “Algorithmische Diskrete Mathematik” is supported by the Deutsche Forschungsgemeinschaft (DFG), grant We 1265/2-1.
    Günter M. Ziegler acknowledges support by a DFG Gerhard-Hess-Forschungsförderungspreis.