Local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. For many classic global properties, like checking the acyclicity of the network, the optimal size of the certificates depends on the size of the network, $n$. In this paper, we focus on properties for which the size of the certificates does not depend on $n$ but on other parameters. We focus on three such important properties and prove tight bounds for all of them. Namely, we prove that the optimal certification size is: $\Theta(\log k)$ for $k$-colorability (and even exactly $\lceil \log k \rceil$ bits in the anonymous model while previous works had only proved a $2$-bit lower bound); $(1/2)\log t+o(\log t)$ for dominating sets at distance $t$ (an unexpected and tighter-than-usual bound) ; and $\Theta(\log \Delta)$ for perfect matching in graphs of maximum degree $\Delta$ (the first non-trivial bound parameterized by $\Delta$). We also prove some surprising upper bounds, for example, certifying the existence of a perfect matching in a planar graph can be done with only two bits. In addition, we explore various specific cases for these properties, in particular improving our understanding of the trade-off between locality of the verification and certificate size.

翻译：本土认证是一种分布式机制，使网络节点能够借助称为证书的少量信息检查当前配置的正确性。对于许多经典全局性质（如检查网络的无环性），证书的最优大小取决于网络规模$n$。本文聚焦于证书大小不依赖于$n$而依赖其他参数的若干重要性质，对其中三种此类性质证明其紧界。具体而言，我们证明：对于$k$-可着色性，最优认证规模为$\Theta(\log k)$（在匿名模型中精确为$\lceil \log k \rceil$比特，而先前工作仅证明了下界为2比特）；对于距离$t$支配集，最优规模为$(1/2)\log t+o(\log t)$（一个出乎意料且比常规更紧的界）；对于最大度为$\Delta$的图的完美匹配，最优规模为$\Theta(\log \Delta)$（首个以$\Delta$为参数的非平凡界）。我们还证明了一些令人惊喜的上界，例如在平面图中仅需两比特即可认证完美匹配的存在性。此外，我们探讨了这些性质的多种特例，特别是深化了对验证局部性与证书规模之间权衡的理解。