We show that the decision problem of recognising whether a triangulated 3-manifold admits a Seifert fibered structure with non-empty boundary is in NP. We also show that the problem of producing Seifert data for a triangulation of such a manifold is in the complexity class FNP. We do this by proving that in any triangulation of a Seifert fibered space with boundary there is both a fundamental horizontal surface of small degree and a complete collection of normal vertical annuli whose total weight is bounded by an exponential in the square of the triangulation size.

翻译：我们证明，判断一个三角化的三维流形是否允许具有非空边界的Seifert纤维结构的判定问题属于NP类。同时证明，为该类流形的三角化生成Seifert数据的问题属于复杂度类FNP。我们通过证明在任意带边界的Seifert纤维空间的三角化中，均存在一个度数较小的基本水平曲面以及一组完整的法向垂直环面，且其总权重受三角化规模平方的指数级边界约束来完成上述证明。