  Finding a cycle of lowest weight that represents a homology class in a
simplicial complex is known as homology localization (HL). Here we address this
NP-complete problem using parameterized complexity theory. We show that it is
W[1]-hard to approximate the HL problem when it is parameterized by solution
size. We have also designed and implemented two algorithms based on treewidth
solving the HL problem in FPT-time. Both algorithms are ETH-tight but our
results shows that one outperforms the other in practice.
