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Given a $0,1$ (binary) integer program of the form:

$$ \begin{array}{lll} \text{min} & f(x) & \\ \text{s.t.} & A x = b \\ & x_i \ge 0 & \quad \forall i\\ & x_i \in \{0,1\} & \quad \forall i \end{array} $$

Note that the size of $A$ is not fixed in either dimension.

I believe this problem has been shown to be hard to approximate (strongly ${\sf NP}$-Complete) by Garey & Johnson. If so, is this still the case when $A, b$ have binary entries and $f(x)$ is a linear function ( $f(x) = \sum_i c_i x_i$ )?

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    $\begingroup$ “Hard to approximate” and “strongly NP-complete” are two different notions. $\endgroup$ – Tsuyoshi Ito Feb 14 '13 at 2:08
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    $\begingroup$ The answer to your question is yes. $\endgroup$ – Chandra Chekuri Feb 14 '13 at 19:35
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One-in-three 3SAT is NP-complete. Looking at the reduction, it inherits the APX-hardness of 3SAT. You can formulate one-in-three 3SAT as a binary integer program with binary entries, so you problem is APX-hard.

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