We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between $\omega(\log^* n)$ and $o(\log n)$. This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where $O(1)$, $\Theta(\log^* n)$, and $\Theta(n)$ are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes.

翻译：我们考虑分布式计算LOCAL模型中的局部可检查标注（LCL）问题。自2016年以来，大量研究工作已系统刻画了LCL问题的可能复杂度谱系。例如，已有结论表明：不存在确定性复杂度介于$\omega(\log^* n)$与$o(\log n)$之间的LCL问题。该研究方向为算法设计者提供了丰富的算法技术和深刻见解。尽管LCL问题在一般图、树和路径上的复杂度图景现已清晰，但这三类图结构之外的图类仍鲜有探索。事实上，近年研究趋势已转向对路径和树领域内特殊实例的精细化分析。本文将该LCL问题复杂度谱系的刻画研究推广至更广泛的图类。我们提出一个猜想，该猜想刻画了任意满足子图闭包性质的图类中LCL问题的复杂度图景，并证明了猜想的部分结论。主要发现如下：1. 我们建立了与路径具有相同确定性复杂度图景的子图闭包图类的简洁刻画——其中仅存在$O(1)$、$\Theta(\log^* n)$和$\Theta(n)$三种复杂度类；2. 自然猜想：当且仅当子图闭包图类具有有界树宽且无界路径宽时，该类与树具有相同的复杂度图景。我们证明了该猜想的必要性方向；3. 除了熟知的路径、树和一般图的复杂度图景外，子图闭包图类中存在无穷多种不同的复杂度图景。