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Let $D$ be a finitely-connected planar domain, or, even more particularly, a domain obtained from the sphere $S^2$ by removing finitely many disjoint open topological disks. Let $\mathrm{PMCG}(D)$ be a pure mapping class group of such a domain $D$, i.e., all its elements fix every boundary curve pointwise.

Is there any (effective) algorithms for solving the conjugacy problem in $\mathrm{PMCG}(D)$? Or, perhaps, for constructing Nierlsen-Thurston decomposition for a given element $\phi \in \mathrm{PMCG}(D)$?

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There are polynomial-time algorithms to solve the conjugacy problem, and also to find the Nielsen-Thurston decomposition, in the usual (non-pure) mapping class group.

It is not immediately clear to me how to reduce the conjugacy problem in the pure mapping class group to the above. If you mostly care about the pseudo-Anosov case, then there is an algorithm (by generating the centraliser in the regular mapping class group of one of the given elements).

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