Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Integrality Number of an Integer Program   We introduce the integrality number of an integer program (IP) in inequality
form. Roughly speaking, the integrality number is the smallest number of
integer constraints needed to solve an IP via a mixed integer (MIP) relaxation.
One notable property of this number is its invariance under unimodular
transformations of the constraint matrix. Considering the largest minor
$\Delta$ of the constraint matrix, our analysis allows us to make statements of
the following form: there exist numbers $\tau(\Delta)$ and $\kappa(\Delta)$
such that an IP with $n\geq \tau(\Delta)$ many variables and $n +
\kappa(\Delta)\cdot \sqrt{n}$ many inequality constraints can be solved via a
MIP relaxation with fewer than $n$ integer constraints. From our results it
follows that IPs defined by only $n$ constraints can be solved via a MIP
relaxation with $O(\sqrt{\Delta})$ many integer constraints.

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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg