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.

翻译：在简单复合物中找到代表同系物类的最小重量周期,称为同系物本地化(HL)。在这里,我们使用参数化复杂度理论来解决这个NP完整的问题。我们表明,当HL问题按溶解大小参数化时,很难接近它。我们还设计并实施了两种基于树宽的算法,在FPT-时间解决HL问题。两种算法都十分严格,但我们的结果表明,一种在实践上优于另一种。