Understanding and generalizing the relaxation-guided round elimination technique

Characterize the theoretical principles that explain why the diagram-driven label-merging relaxation procedure used to construct round-elimination sequences yields correct lower bounds, and determine the breadth of locally checkable labeling (LCL) problems for which this technique applies.

Background

To apply round elimination, the authors introduce a mechanical rule based on diagrams of label strengths and targeted label mergers. Surprisingly, qualitatively similar relaxations work for both maximal matching and the newly introduced iterated GHZ problem, despite their apparent differences.

While this heuristic produced the desired lower bounds and replicated prior results, the authors explicitly state that the underlying reason for its success and its scope of applicability are unclear, leaving a conceptual and methodological gap that invites further investigation.