  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.
