some fundamental differences when dealing with vertex-certifiable or packing-certifiable problems.

Indeed, as discussed in more detail in Section \[sec:generic\], with an encoding ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ we associate a function $f^{{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ that corresponds, roughly speaking, to the maximum size of a partial solution that satisfies the constraints defined by ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$. In order for an encoder to be $g$-confined for some function $g(t)$ of the treewidth $t$, for some vertex-certifiable problems such as $r$-<span style="font-variant:small-caps;">Scattered Set</span> (see [@KviaDP]) we need to “force” the confinement artificially, in the sense that we directly discard the entries in the tables whose associated values differ by more than $g(t)$ from the maximum (or minimum) ones. Fortunately, we can prove that an encoder with this modified function is still DP-friendly. However, this is not the case for packing-certifiable problems such as <span style="font-variant:small-caps;">$\mathcal{F}$-Packing</span>. Intuitively, the difference lies on the fact that in a packing-certifiable problem, a solution of size $k$ can contain arbitrarily many vertices (for instance, if one wants to find $k$ disjoint cycles in an $n$-vertex graph with girth $\Omega(\log n)$) and so it can as well contain arbitrarily many vertices from any subgraph corresponding to a rooted subtree of a tree decomposition of the input graph 