I have the following problem:

$A x = b$

where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly).

$x \in \{0,1\}^n $ - vector of binary variables, **which need to be found**.

Investigation shows it's known **NP-complete** problem. But special case of this problem ($m = 1$) is Subset sum problem, which also is NP-complete but has good pseudo-polynomial dynamic programming solution. For my problem coefficients in $A, b$ are bounded and not very big ($A_{i,j}, b_i \leq S $), but $m$ can be quite large, so it will be unacceptable to have $O(n \times S^m)$ solution. But may be there is something better? Optionally, if it's significantly easier, **method which gives only one solution instead of all, also will be helpful**.

**Could someone point me to good method for solving such kind of problems?**