The last decade of research on the LOCAL model has seen tremendous progress in understanding locally checkable labeling (LCL) problems, culminating in an almost complete classification of the possible complexities LCL problems can exhibit. In particular, on undirected trees, Chang and Pettie showed that there is no LCL problem with complexity between $ω(\log n)$ and $n^{o(1)}$ and Chang showed that, for every positive integer $k$, there is no LCL problem with complexity between $ω(n^{1/(k+1)})$ and $o(n^{1/k})$; additionally, which side of each gap a problem is found on is decidable. While the class of LCL problems - which, roughly speaking, consists of problems for which the correctness of a solution can be described by a finite set of allowed node configurations, which in turn can be locally verified by a constant-time algorithm - includes many important problems, it has one major restriction: problems can be defined only on bounded degree graphs, which consequently restricts all the classification and gap results mentioned above. In this work, we propose a generalization of LCL problems to unbounded degree using Presburger monadic second-order (PMSO) formulas; more specifically, we consider what we call Local PMSO (LPMSO) problems, i.e., problems whose correct solutions are both finitely described by a PMSO formula and locally verifiable by a LOCAL algorithm in constant time - this class contains many of the important problems studied in the LOCAL model but defines them on unbounded degree graphs. As our main result we prove that, on unbounded degree rooted trees, the aforementioned $ω(\log n)$ - $n^{o(1)}$ and $ω(n^{1/(k+1)})$ - $o(n^{1/k})$ complexity gaps (and their decidability) extend to the class of LPMSO problems.

翻译：过去十年间，关于LOCAL模型的研究在理解局部可检查标记（LCL）问题上取得了巨大进展，最终几乎完整分类了LCL问题可能呈现的复杂度。特别是在无向树上，Chang和Pettie证明了不存在复杂度介于$ω(\log n)$与$n^{o(1)}$之间的LCL问题，而Chang进一步证明：对任意正整数$k$，不存在复杂度介于$ω(n^{1/(k+1)})$与$o(n^{1/k})$之间的LCL问题；此外，每个问题位于上述区间的哪一侧是可判定的。尽管LCL问题类（大致而言，其解的正确性可由有限允许节点配置集合描述，且可通过常数时间算法局部验证）包含许多重要问题，但它有一个主要限制：问题只能定义在有界度图上，这进而限制了上述所有分类与间隙结果。本文中，我们利用Presburger一元二阶（PMSO）公式，将LCL问题推广至无界度图；具体而言，我们研究所谓局部PMSO（LPMSO）问题——其正确解既可由PMSO公式有限描述，又可通过常数时间的LOCAL算法局部验证——此类问题包含LOCAL模型中研究的许多重要问题，但将其定义在无界度图上。作为主要结果，我们证明：在无界度有根树上，前述$ω(\log n)$ - $n^{o(1)}$与$ω(n^{1/(k+1)})$ - $o(n^{1/k})$复杂度间隙（及其可判定性）可扩展至LPMSO问题类。