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My question concerns the NP-hardness of the following discrete optimization problem:

Given a matrix $A \in \{ \pm 1 \}^{m\times n}$,

$$\begin{array}{ll} \underset{x \in \{ \pm 1 \}^m ,\, y \in \{ \pm 1 \}^n}{\text{maximize}} & x^T A \, y\end{array}$$

Is this problem known to be NP-hard?

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  • $\begingroup$ If $A$ can also have zero entries, then it is NP-hard. I doubt this is really necessary, but I don't have an immediate argument for this. $\endgroup$ Commented Jun 23, 2021 at 19:24

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NP-hardness is proved by Roth and Viswanathan in the paper On the hardness of decoding the gale-berlekamp code

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