Locally checkable labeling problems (LCLs) form the foundation of the modern theory of distributed graph algorithms. First introduced in the seminal paper by Naor and Stockmeyer [STOC 1993], these are graph problems that can be described by listing a finite set of valid local neighborhoods. This seemingly simple definition strikes a careful balance between two objectives: they are a family of problems that is broad enough so that it captures numerous problems that are of interest to researchers working in this field, yet restrictive enough so that it is possible to prove strong theorems that hold for all LCL problems. In particular, the distributed complexity landscape of LCL problems is now very well understood. In this work we show that the family of LCL problems is extremely robust to variations. We present a very restricted family of locally checkable problems (essentially, the "node-edge checkable" formalism familiar from round elimination, restricted to regular unlabeled graphs); most importantly, such problems cannot directly refer to e.g. the existence of short cycles. We show that one can translate between the two formalisms (there are local reductions in both directions that only need access to a symmetry-breaking oracle, and hence the overhead is at most an additive $O(\log^* n)$ rounds in the LOCAL model).

翻译：局部可检查标记问题（LCLs）构成了分布式图算法现代理论的基础。这类问题最初由Naor和Stockmeyer的开创性论文[STOC 1993]提出，可通过列举有限个有效局部邻域来描述图问题。这个看似简单的定义在两个目标之间实现了精妙的平衡：该问题族足够广泛，能够涵盖该领域研究者关注的众多问题；同时又具有足够的限制性，使得能够证明对所有LCL问题都成立的强定理。特别地，目前对LCL问题的分布式计算复杂性图景已有非常深入的理解。本研究表明，LCL问题族对形式定义的变体具有极强的鲁棒性。我们提出一个高度受限的局部可检查问题族（本质上是轮次消去法中常见的“节点-边可检查”形式化描述，且限制在正则无标记图上）；最关键的是，这类问题无法直接引用诸如短环存在性等结构特征。我们证明可以在两种形式化体系间进行转换（存在双向局部规约，仅需借助对称性破缺预言机，因此在LOCAL模型中的开销至多为附加的$O(\log^* n)$轮次）。