Testing additive integrality gaps

Authors:
Friedrich Eisenbrand;Nicolai Hähnle;Dömötör Palvolgyi;Gennady Shmonin
Affiliations:
Ecole Polytechnique Fédérale de Lausanne, Switzerland;Ecole Polytechnique Fédérale de Lausanne, Switzerland;Ecole Polytechnique Fédérale de Lausanne, Switzerland;Ecole Polytechnique Fédérale de Lausanne, Switzerland
Venue:
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Year:
2010

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Abstract

We consider the problem of testing whether the maximum additive integrality gap of a family of integer programs in standard form is bounded by a given constant. This can be viewed as a generalization of the integer rounding property, which can be tested in polynomial time if the number of constraints is fixed. It turns out that this generalization is NP-hard even if the number of constraints is fixed. However, if, in addition, the objective is the all-one vector, then one can test in polynomial time whether the additive gap is bounded by a constant.