A parameter $\kappa'$ of a problem is dominated by a parameter $\kappa$ if $\kappa'(x, D) \le g(\kappa(x, D))$ for a function $g$ and every instance $x$ and threshold $D$. Then the identity on words is an fpt-reduction from the problem with $\kappa$ to the problem with $\kappa'$ when $g$ is computable, and the words with $\kappa \le k$ are among those with $\kappa' \le g(0) + \dots + g(k)$.

Theorem. If $\kappa'$ is dominated by $\kappa$, then para-NP-hardness and W[t]-hardness for $\kappa$ hold for $\kappa'$, and membership in XP and fixed-parameter tractability for $\kappa'$ hold for $\kappa$ ($g$ computable for the last two).

Two dominations of the zoo's parameters: a summand is dominated by a sum, and on the operations family the number $\#p$ of distinct processing times is at most $p_{\max} + 1$ (the values lie in $\{0, \dots, p_{\max}\}$).

Theorem. These dominations hold.