The Waring rank of a degree-$d$ homogeneous polynomial $f$ is the minimum number of linear forms whose sum of $d$th powers is equal to $f$. For nonnegative integers $n$ and $d$, where $n \ge d$, define $A(n,d)$ as the minimum Waring rank of a degree-$d$ polynomial in $n$ variables that is supported exactly on the set of all degree-$d$ squarefree monomials. We show that this and related quantities have intimate connections to the areas of parameterized and exact algorithms, generalizing and improving previous methods and providing a concrete approach to obtain faster approximate counting and deterministic decision algorithms.