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[Solved]: Hardness of Approximating 0-1 Integer Programs

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Problem Detail: 

Given a $0,1$ (binary) integer program of the form: $$ \begin{array}{lll} \text{min} & f(x) & \\ \text{s.t.} &A\vec{x} = \vec{b} & \quad \forall i\\ &x_i\ge 0 & \quad \forall i\\ &x_i \in \{0,1\} & \quad \forall i \end{array} $$

Note: 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) Garey & Johnson.

If so, is this still the case when $A$, $\vec{b}$ have binary entries and $f(x)$ is a linear function ( $f(x) = \sum_i c_i x_i$ )?

Asked By : Jonas Anderson

Answered By : Yuval Filmus

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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