Additional Efficiently Solvable LLL Applications

Identify further concrete applications, including vertex coloring and degree splitting problems, whose distributed Lovsz Local Lemma instances can be solved efficiently, particularly when their bad events are concentration failures with slack in the allowable error.

Background

The paper demonstrates that specially structured LLL instances arising from triangle-free and girth-5 coloring can be solved in polylogarithmic-in-logarithm time through resilience. This motivates the search for other algorithmic applications whose LLL instances possess comparable concentration and slack properties.

The authors specifically identify vertex coloring and degree splitting as possible domains for finding additional efficiently solvable instances, while noting that the broader general LLL problem may remain out of reach.

References

If no efficient general LLL algorithm can be found, can we further identify applications with efficiently-solvable LLL instances, such as in other vertex coloring or degree splitting problems?

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

Molloy showed that the constant in the number of colors needed for coloring triangle-free coloring can be improved from $4$ to $1$, but no (nontrivial) distributed algorithm yet exists to match this bound. Can such an algorithm be devised, and if so (since it will almost certinly involve LLL calls as in ) can the LLL instances therein be efficiently solved, similarly to this work?

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