# Cyclic group and knapsack facets

@article{Aroz2003CyclicGA, title={Cyclic group and knapsack facets}, author={Juli{\'a}n Ar{\'a}oz and Lisa Evans and Ralph E. Gomory and Ellis L. Johnson}, journal={Mathematical Programming}, year={2003}, volume={96}, pages={377-408} }

Abstract. Any integer program may be relaxed to a group problem. We define the master cyclic group problem and several master knapsack problems, show the relationship between the problems, and give several classes of facet-defining inequalities for each problem, as well as a set of mappings that take facets from one type of master polyhedra to another.

## 54 Citations

### Generating facets for finite master cyclic group polyhedra using n-step mixed integer rounding functions

- MathematicsEur. J. Oper. Res.
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### A Polyhedral Study of the Mixed Integer Cut

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The Gomory mixed integer cut is studied as a facet of the master cyclic group polyhedron and its extreme points and adjacent facets in this setting are characterized.

### Corner Polyhedra and their connection with cutting planes

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Experiments are described that indicate the dominance of a relatively small number of the facets of Corner Polyhedra, which has implications for their value as cutting planes.

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The lower bound value obtained when adding (implicitly) all the interpolated subadditive cuts that can be derived from the individual rows of an optimal LP tableau is computed, thus approximating the optimization over the so-called Gomory's Corner polyhedron.

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A minimal representation of the subadditive polytope for themaster cyclic group problem is discovered and a max °ow model on coveringspace is shown to be equivalent to the dual of shooting linear programming problem.

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An integer version of the cover inequalities are given and a necessary and sufficient facet condition for them are described, which generalizes the well-known facet condition of minimality of covers for 0-1 knapsacks.

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This paper numerically shows that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.

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An explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP is presented, which gives a polynomial time algorithm for separating an arbitrary point from MEP.

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