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 ( $f(x) = \sum_i c_i x_i$ )?
Hardness of approximating 0-1 integer programs
complexity theory;np complete;approximation;integer programming
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.