Regression/compressive sensing with non-linear constrains where the coefficients are assumed to be integer or binary {0,1}

Question

The following regression problem $$ \mathbf{y} = \mathbf{A}\mathbf{x} $$ where $\mathbf{y}$ is a $N\times 1$ column real vector, $\mathbf{A}$ is a $N\times M$ real matrix where each column $\mathbf{a}_i$ can be called an atom, $N$ could be great than $M$, or $N=M$, or $N>M$. We want to solve $\mathbf{x}$ with minimal loss where

1. each entry of $\mathbf{x}$ is integers, i.e., $x_i=1,2,...,\infty$, or
2. each entry of $\mathbf{x}$ is either $0$ or $1$, $x_i=\{0,1\}$

How could I solve this problem ? any hint on existing implementation code or where could I find references on this field ?