Precise running time of the algorithmic Lovasz Local Lemma application

Determine a precise polynomial running-time bound for the algorithmic Lovász Local Lemma version used to obtain the multipartite independent sets required by the bramble-construction algorithm.

Background

The paper claims polynomial-time construction of the brambles when the relevant path system or linked path is provided. Its main algorithmic bottleneck is the application of the algorithmic Lovász Local Lemma, specifically the version attributed to Harris. While that result guarantees polynomial time, the paper states that the degree of the polynomial is not specified, preventing a precise running-time bound for the overall construction.

References

Although this allows us to nullify any change to the statement of {thm:main-proof} due to the Lovász Local Lemma, we remark that a precise running time on {prop:independent-set-multipartite} is not known, and thus we can only claim that with {thm:main-proof} one can build the bramble in polynomial time.

— Polynomial directed treewidth forces a large bramble with congestion two  (2610.13270 - Campos et al., 9 Oct 2026) in Section 5.1, paragraph “Finding the brambles in polynomial time”