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

Abstract

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 x^new ∈ $\mathcal{F}$with cx ^new > c x ⁰ such that x ^new−x 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.