I have a hard time with integer programming, so I'm not going to say anything about it.  My understanding of it is that the input is a system of polynomials with variables that have unknown values.  Each variable can be only 0 or 1.

Karp's list of NP-complete problems mentioned a matrix and a vector.  I think he was talking about the same thing, but I'm not sure.

Maybe someone else with a better grasp of this problem will explain it.