Generating functions and duality for integer programs

Generating functions and duality for integer programs

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Article ID: iaor20051099
Country: Netherlands
Volume: 1
Issue: 2
Start Page Number: 167
End Page Number: 187
Publication Date: Nov 2004
Journal: Discrete Optimization
Authors:
Abstract:

We consider the integer program: max c′x|Ax=y;xNn. Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables yRm have discrete analogues zCm, related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P*, the analogue of the dual in linear programming.

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