is contained in the root bag. In the following we always consider boundaried nice tree decompositions of width $t-1$, which can be computed in polynomial time for fixed $t$ [@Klo94; @Bod96].

A framework to replace protrusions for packing problems {#sec:generic}
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In this section we restate and in many cases modify the definitions given in [@KviaDP] in order to deal with packing-certifiable problems; we will point out the differences. As announced in the introduction, missing proofs can be found in Appendix \[ap:framework\].

**Encoders.** In the following we extend the definition of an encoder given in [@KviaDP Definition 3.2] so that it is able to deal with packing-certifiable problems. The main difference is that now the function ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}$ is incorporated in the definition of an encoder, since as discussed in the introduction we need to consider an additional scenario where the entries of the table are discarded (technically, this is modeled by setting those entries to “$-\infty$”) and for this we will have to deal with the partial solutions particular to each problem. In the applications of the next sections, we will call such functions that propagate the entries to be discarded *relevant*. We also need to add a condition about 