What is the decision version of integer programming?

I don't know what is meant by "decision version of integer programming".

I know ILP, but this terminology has me confused.

It's the same as for all NP problems; the optimisation problem is

Find a valid solution $s$ that minimises¹ $f(s)$!

and the corresponding decision problem is

Is there a valid solution $s$ with $f(s) \leq k$?

You see that the former immediately solves the latter, and you can solve the former by using the latter with binary search over the set of feasible $k$.

• Of course, you can have "maximise" here and "$\geq k$" in the decision version.

StackExchange Computer Science Q#32624, answer score: 8