Locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer, are the standard formalism for studying local distributed graph problems. They capture many natural problems, such as coloring, maximal independent set, and sinkless orientation, while still being restrictive enough to enable general complexity-theoretic results. However, recent work has also revealed artificial LCLs with counterintuitive behavior, including quantum and shared-randomness advantages, exotic round complexities, dependence on computability assumptions, and undecidability phenomena. This raises a natural question: are these phenomena artifacts of the particular Naor-Stockmeyer definition, or are they inherent to local checkability?

We show that there is a one-round randomized distributed algorithm that can 2-color cycles such that the expected fraction of monochromatic edges is less than 0.24118. We also show that a one-round algorithm cannot achieve a fraction less than 0.23879. Before this work, the best upper and lower bounds were 0.25 and 0.2. Our proof was largely discovered and developed by large language models, and both the upper and lower bounds have been formalized in Lean 4.

Machine learning is increasingly used in the real world, including in sensitive contexts. This, combined with their innate non-transparency, gives rise to the need for explaining the decisions of machine learning models. We focus on optimal counterfactual explanations for random forests, a well-performing and popular classifier, intuitively answering the question “What is the cheapest way to change a given classification?” We propose an algorithm based on state-of-the-art maximum satisfiability (MaxSAT) solving and a compact and faithful formal model of the classifier. An optimal counterfactual is guaranteed to be found for any sample, and further plausibility constraints (e.g. immutability of some features) as well as custom cost function can be seamlessly incorporated. We conduct an empirical evaluation showing promising run time performance for our approach compared to existing optimal algorithms for computing counterfactuals for random forests, outperforming previous approaches on many datasets.