We provide a $O(\log^6 \log n)$-round randomized algorithm for distance-2 coloring in CONGEST with $\Delta^2+1$ colors. For $\Delta\gg\operatorname{poly}\log n$, this improves exponentially on the $O(\log\Delta+\operatorname{poly}\log\log n)$ algorithm of [Halld\'orsson, Kuhn, Maus, Nolin, DISC'20]. Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.

翻译：我们提出了一个在CONGEST模型下使用Δ²+1种颜色进行距离-2着色的O(log⁶ log n)轮随机算法。对于Δ≫poly log n的情形，该算法在效率上指数级优于[Halldórsson, Kuhn, Maus, Nolin, DISC'20]中提出的O(log Δ+poly log log n)算法。本研究源于CONGEST模型中局部约简的普遍性与困难性。例如，局部引理[Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23]算法通常假定在冲突图上进行通信，这在LOCAL模型中仅需常数开销即可模拟，而在CONGEST中代价可能高得难以承受。我们希望所提出的技术有助于解决CONGEST模型中由局部关系定义的其他着色问题。