Is 0-1 programming with constant number of constraints polynomially solvable?
I'm assuming that by "0-1 programming with a constant number of constraints" you mean the following problem:

Maximize some linear function of (x_1, x_2, ..., x_n) subject to the constraints that each x_i is in {0,1} and a constant number of additional linear constraints.

This problem is NP-complete even with 1 additional constraint since 0-1 knapsack can be written in this form.