In this work we introduce the graph-theoretic notion of mendability: for each locally checkable graph problem we can define its mending radius, which captures the idea of how far one needs to modify a partial solution in order to "patch a hole." We explore how mendability is connected to the existence of efficient algorithms, especially in distributed, parallel, and fault-tolerant settings. It is easy to see that $O(1)$-mendable problems are also solvable in $O(\log^* n)$ rounds in the LOCAL model of distributed computing. One of the surprises is that in paths and cycles, a converse also holds in the following sense: if a problem $\Pi$ can be solved in $O(\log^* n)$, there is always a restriction $\Pi' \subseteq \Pi$ that is still efficiently solvable but that is also $O(1)$-mendable. We also explore the structure of the landscape of mendability. For example, we show that in trees, the mending radius of any locally checkable problem is $O(1)$, $\Theta(\log n)$, or $\Theta(n)$, while in general graphs the structure is much more diverse.

中文翻译：

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本地修补

在这项工作中，我们介绍了可修复性的图论概念：对于每个局部可检查的图问题，我们都可以定义其修补半径，这捕捉了一个想法，即为了“修补漏洞”，需要修改部分解的程度。我们探索可修复性如何与有效算法的存在联系起来，特别是在分布式，并行和容错设置中。很容易看出，在分布式计算的本地模型中，在$ O（\ log ^ * n）$轮次中，$ O（1）$可解决的问题也可以解决。令人惊讶的是，在路径和循环中，反过来也具有以下含义：如果可以在$ O（\ log ^ * n）$中解决问题$ \ Pi $，则总会有一个限制\\ Pi '\ subseteq \ Pi $仍然可以有效解决，但也可以修改$ O（1）$。我们还探索了可修复性景观的结构。例如，我们显示在树中，任何局部可检查问题的修补半径为$ O（1）$，$ \ Theta（\ log n）$或$ \ Theta（n）$，而在一般图中，结构更加多样化。