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Suppose I have a linear system $$ A_{ij}X_j = B_i $$

where $A$ is a $m\times n$ of real values and $B$ is a vector of real values. When $X$ is allowed to be real, then a solution of this system for X exists if $A$ is invertible (which requires $A$ to be square and have at least the same dimension as $X$).

What if each element of the vector $X$ is only allowed to take the values 0 or 1? Is there a general procedure for solving for $X$? Can one make a general statement of how many linear equations are needed to find a solution?

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    I think this is NP-hard in general. Even when $A$ has only one row $(a_1,\ldots,a_n)$ and the rest are $0$, this is knapsack problem. See [here](http://lara.epfl.ch/w/_media/papadimitriou81complexityintegerprogramming.pdf) for more details. – 2017-01-19

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This is an integer program, which is typically NP-hard. In some cases (when the integrality gap is zero), then we can relax the $x_j$ to be real valued, solve a linear program, and round the result to get the exact result. In general solving the LP-relaxation will not give you the correct solution.