Finding the smallest $d$-chain with a specific $(d-1)$-boundary in a simplicial complex is known as the \textsc{Minimum Bounded Chain} (MBC$_d$) problem. The MBC$_d$ problem is NP-hard for all $d\geq 2$. In this paper, we prove that it is also W[1]-hard for all $d\geq 2$, if we parameterize the problem by solution size. We also give an algorithm solving the MBC$_1$ problem in polynomial time and introduce and implemented two fixed parameter tractable (FPT) algorithms solving the MBC$_d$ problem for all $d$. The first algorithm is a generalized version of Dijkstra's algorithm and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

翻译：在简单复合物中找到一个特定美元(d-1)美元边界线的最小的美元链,称为 \ textsc{ minBounded chail} (MBC$_d$) 问题。 MBC$_d$问题是所有$d\geq 2$的硬NP。在本文中,我们证明,如果我们用解决方案大小来参数化问题,那么所有$d\geq 2$也是W[1]-hard。我们还给出一个算法,在多元时间内解决MBC$_1美元的问题,并引入和实施两种固定参数可移动的算法(FPT),解决所有美元 MBC$_d$的问题。第一个算法是Dijkstra的算法的通用版本,并按解算法大小和共同面度进行参数比较。第二个算法是一种动态的编程方法,以树线线为基础,其运行时间与我们在指数时间假设下证明的较低约束时间相同。