I'm aware that 0-1 integer programming problem is NP-complete, where the problem is stated as: Given some integer matrix A and some integer vector b, determine whether there exists a vector x consisting of 0's and 1's such that Ax >= b. I've seen that 3-CNF SAT is reducible to this problem.
However, here's a slight variant: Given some integer matrix A and some integer vector b, determine whether there exists a vector x consisting of 0's and 2's such that Ax >= b.
The fact that vector x must consist of 0's and 2's kind of throws me off here. For the original problem, we just could reduce from 3-SAT by adding the inequalities 0 <= xi <= 1 for each boolean variable. But here, I can't just add the following inequalities:
0 <= xi <= 2
because x cannot equal 1.
Am I allowed to add an inequality, such as xi != 1? If not, what reduction can I use to show that the given problem is NP-complete? Thanks for any help.