We present an algorithm for the following problem. Given a triangulated 3-manifold M and a (possibly non-simple) closed curve on the boundary of M, decide whether this curve is contractible in M. Our algorithm runs in space polynomial in the size of the input, and (thus) in exponential time. This is the first algorithm that is specifically designed for this problem; it considerably improves upon the existing bounds implicit in the literature for the more general problem of contractibility of closed curves in a 3-manifold. The proof of the correctness of the algorithm relies on methods of 3-manifold topology and in particular on those used in the proof of the Loop Theorem. As a byproduct, we obtain an algorithmic version of the Loop Theorem that runs in polynomial space, and (thus) in exponential time.

中文翻译：

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确定 3 流形边界上非简单曲线的可收缩性：计算循环定理

我们提出了一个算法来解决以下问题。给定一个三角化的 3 流形 M 和 M 边界上的（可能是非简单的）闭合曲线，确定这条曲线在 M 中是否可收缩。我们的算法在输入大小的空间多项式中运行，并且（因此）在指数时间。这是第一个专门为这个问题设计的算法；它大大改进了文献中隐含的现有界限，用于解决更一般的 3 流形中闭合曲线的可收缩性问题。算法正确性的证明依赖于 3-流形拓扑的方法，特别是循环定理证明中使用的方法。作为副产品，我们获得了循环定理的算法版本，它在多项式空间中运行，并且（因此）在指数时间内运行。