Polylogarithmic Distributed Algorithms for the General LLL

Establish a log^{O(1)}\log n-round distributed algorithm that solves all Lovsz Local Lemma instances, potentially under polynomially weakened criteria, or prove an improved lower bound beyond the existing \Omega(\log\log n) rounds.

Background

The paper uses resilient Lovsz Local Lemma instances to obtain \log{O(1)}\log n-round algorithms for difficult coloring-related instances. It places this progress against the broader complexity of the general distributed LLL, for which an exponential gap remains between known upper and lower bounds.

The unresolved issue is whether the techniques can be extended to every LLL instance, perhaps under the polynomially weakened criteria used in recent work. The alternative challenge is to improve the known \Omega(\log\log n)-round lower bound if such a general algorithm cannot be obtained.

References

Despite recent progress, an exponential gap still remains in the complexity of the general distributed LLL. Can a \log{O(1)}\log n round algorithm that solves all LLL instances (perhaps with polynomially weakened criteria, as in , or can the \Omega(\log\log n)-round lower bound of be improved?

— Triangle-Free Coloring in LOCAL via Resilient Lovász Local Lemma  (2608.13357 - Davies-Peck et al., 13 Aug 2026) in Section 6, subsection "General LLL"