Figure 10: The green and blue rings intersect, which can create cycles in $R$ when replacing paths.
Pushing a Solution Onto $R$. So far, we have argued that if there is a solution, then there is also one such that the sum of the potential of all of the groups of all of its paths is at most $2^{\mathcal{O}(k)}$. Additionally, we discussed the intuition why this, in turn, implies the following result.
Lemma 2.4. If there is a solution $\mathcal{P}$, then there is a weak linkage pushed onto $R$ that is discretely homotopic to $\mathcal{P}$ and uses at most $2^{\mathcal{O}(k)}$ copies of every edge.
The formal proof of Lemma 2.4 (in Section 8) is quite technical. On a high level, it consists of three phases. First, we push onto $R$ all sequences of the solution—that is, maximal subpaths that touch (but not necessarily cross) $R$ only at their endpoints. Second, we eliminate some U-turns of the resulting weak linkage (see Fig. 11), as well as “move through” $R$ segments with both endpoints being internal vertices of the same maximal degree-2 path of $R$ and crossing it in opposing directions (called swollen segments). At this point, we are able to bound by $2^{\mathcal{O}(k)}$ the number of segments of the pushed weak linkage. Third, we eliminate all of the remaining U-turns, and show that then, the number of copies of each edge used must be at most $2^{\mathcal{O}(k)}$. We also modify the pushed weak linkage to be of a certain “canonical form” (see Section 8).
Generating a Collection of Pushed Weak Linkages. In light of Lemma 2.4 and Proposition 2.1, it only remains to generate a collection of $2^{\mathcal{O}(k^2)}$ pushed weak linkages that includes all pushed weak linkages (of some canonical form) using at most $2^{\mathcal{O}(k)}$ copies of each edge. (This part, along with the preprocessing and construction of $R$, are the algorithmic parts of our proof.)
This part of our proof is essentially a technical modification and adaptation of the work of Schrijver [42] (though we need to be more careful to obtain the bound $2^{\mathcal{O}(k^2)}$). Thus, we only give a brief description of it in the overview. Essentially, we generate pairs of a pairing and a template: a pairing assigns, to each vertex $v$ of $R$ of degree 1 or at least 3, a set of pairs of edges incident to $v$ to indicate that copies of these edges are to be visited consecutively (by at least one walk of the weak linkage under construction); a template further specifies, for each of the aforementioned pairs of edges, how many times copies of these edges are to be visited consecutively (but not which copies are paired-up). Clearly, there is a natural association of a pairing and a template to a pushed weak linkage. Further, we show that to generate all pairs of pairings and templates associated with the weak linkages we are interested in, we only need to consider pairings that in total have $\mathcal{O}(k)$ pairs and templates that assign numbers bounded by $2^{\mathcal{O}(k)}$ (because we deal with weak linkages using $2^{\mathcal{O}(k)}$ copies of each edge):
Lemma 2.5. There is a collection of $2^{\mathcal{O}(k^2)}$ pairs of pairings and templates that, for any canonical pushed weak linkage $\mathcal{W}$ using only $2^{\mathcal{O}(k)}$ copies of each edge, contains a pair (of a pairing and a template) “compatible” with $\mathcal{W}$. Further, such a collection is computable in time $2^{\mathcal{O}(k^2)}$.
Using somewhat more involved arguments (in Section 9), we also prove the following.
Lemma 2.6. Any canonical pushed weak linkage is “compatible” with exactly one pair of a pairing and a template. Moreover, given a pair of a pairing and a template, if a canonical pushed weak linkage compatible with it exists, then it can be found in time polynomial in its size.