How to find minimum number of k-input LUTs needed to express a n variable boolean function?
The following decision version of your problem is NP-complete:


  Given a formula $f$, an integer $k$, and an integer $s$ (encoded in unary), determine whether $f$ can be represented using at most $s$ many $k$-input LUT's.


Indeed, you can easily solve SAT by asking whether the instance of SAT can be represented using at most one 0-input LUT. If you disallow $k=0$, you need to work a bit harder: a formula $\phi$ is satisfiable iff $\phi \oplus x$ (where $x$ is a fresh variable) can be represented using at most one 1-input LUT.