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For a fixed marked surface $S$, we construct polynomial bounds on theperiodic and preperiodic lengths of the maximal splitting sequences of aprojectively invariant measured train track.  We give two consequences of these bounds. Firstly, that the problem ofdeciding whether a mapping class is pseudo-Anosov lies in $\textbf{NP}$. Thisis dual to the previously known result that the pseudo-Anosov problem is in$\textbf{co-NP}$. Secondly, that the problem of deciding whether two mappingclasses are conjugate lies in $\textbf{co-NP}$. Similarly, this is the dual tothe previously known result that the conjugacy problem is in $\textbf{NP}$.  As usual, in both cases we immediately obtain exponential time solutions tothese problems. A version of these algorithms have been implemented as part offlipper.

The pseudo-Anosov and conjugacy problems are in $\textbf{NP} \cap \textbf{co-NP}$