Recent research revealed the existence of gaps in the complexity landscape of locally checkable labeling (LCL) problems in the LOCAL model of distributed computing. For example, the deterministic round complexity of any LCL problem on bounded-degree graphs is either $O(\log^\ast n)$ or $\Omega(\log n)$ [Chang, Kopelowitz, and Pettie, FOCS 2016]. The complexity landscape of LCL problems is now quite well-understood, but a few questions remain open. For bounded-degree trees, there is an LCL problem with round complexity $\Theta(n^{1/k})$ for each positive integer $k$ [Chang and Pettie, FOCS 2017]. It is conjectured that no LCL problem has round complexity $o(n^{1/(k-1)})$ and $\omega(n^{1/k})$ on bounded-degree trees. As of now, only the case of $k = 2$ has been proved [Balliu et al., DISC 2018]. In this paper, we show that for LCL problems on bounded-degree trees, there is indeed a gap between $\Theta(n^{1/(k-1)})$ and $\Theta(n^{1/k})$ for each $k \geq 2$. Our proof is constructive in the sense that it offers a sequential algorithm that decides which side of the gap a given LCL problem belongs to. We also show that it is EXPTIME-hard to distinguish between $\Theta(1)$-round and $\Theta(n)$-round LCL problems on bounded-degree trees. This improves upon a previous PSPACE-hardness result [Balliu et al., PODC 2019].

中文翻译：

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树上分布式局部可检查问题的复杂性

最近的研究表明，在分布式计算的 LOCAL 模型中，本地可检查标签 (LCL) 问题的复杂性领域存在差距。例如，有界度图上任何 LCL 问题的确定性轮复杂度要么是 $O(\log^\ast n)$ 要么是 $\Omega(\log n)$ [Chang、Kopelowitz 和 Pettie，FOCS 2016] . LCL 问题的复杂性现在已经很清楚了，但仍有一些问题尚待解决。对于有界度树，对于每个正整数 $k$ [Chang and Pettie, FOCS 2017] 存在轮复杂度 $\Theta(n^{1/k})$ 的 LCL 问题。推测在有界度树上没有 LCL 问题具有圆复杂度 $o(n^{1/(k-1)})$ 和 $\omega(n^{1/k})$。截至目前，仅证明了 $k = 2$ 的情况 [Balliu 等人，DISC 2018]。在本文中，我们表明，对于有界树上的 LCL 问题，$\Theta(n^{1/(k-1)})$ 和 $\Theta(n^{1/k})$ 之间确实存在差距每个 $k \geq 2$。我们的证明是建设性的，因为它提供了一种顺序算法，可以决定给定 LCL 问题属于差距的哪一侧。我们还表明，在有界度树上区分 $\Theta(1)$-round 和 $\Theta(n)$-round LCL 问题是 EXPTIME 困难的。这改进了之前的 PSPACE 硬度结果 [Balliu 等人，PODC 2019]。我们还表明，在有界度树上区分 $\Theta(1)$-round 和 $\Theta(n)$-round LCL 问题是 EXPTIME 困难的。这改进了之前的 PSPACE 硬度结果 [Balliu 等人，PODC 2019]。我们还表明，在有界度树上区分 $\Theta(1)$-round 和 $\Theta(n)$-round LCL 问题是 EXPTIME 困难的。这改进了之前的 PSPACE 硬度结果 [Balliu 等人，PODC 2019]。