We consider the problem of testing whether the maximum 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.

Published in:
21st ACM-SIAM Symposium on Discrete Algorithms (SODA10), 1227-1234
Presented at:
21st ACM-SIAM Symposium on Discrete Algorithms, Austin, Texas, January 17-19, 2010
Year:
2010
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