It is known that, for every $k\geq 2$, $C_{2k}$-freeness can be decided by a generic Monte-Carlo algorithm running in $n^{1-1/\Theta(k^2)}$ rounds in the CONGEST model. For $2\leq k\leq 5$, faster Monte-Carlo algorithms do exist, running in $O(n^{1-1/k})$ rounds, based on upper bounding the number of messages to be forwarded, and aborting search sub-routines for which this number exceeds certain thresholds. We investigate the possible extension of these threshold-based algorithms, for the detection of larger cycles. We first show that, for every $k\geq 6$, there exists an infinite family of graphs containing a $2k$-cycle for which any threshold-based algorithm fails to detect that cycle. Hence, in particular, neither $C_{12}$-freeness nor $C_{14}$-freeness can be decided by threshold-based algorithms. Nevertheless, we show that $\{C_{12},C_{14}\}$-freeness can still be decided by a threshold-based algorithm, running in $O(n^{1-1/7})= O(n^{0.857\dots})$ rounds, which is faster than using the generic algorithm, which would run in $O(n^{1-1/22})\simeq O(n^{0.954\dots})$ rounds. Moreover, we exhibit an infinite collection of families of cycles such that threshold-based algorithms can decide $\mathcal{F}$-freeness for every $\mathcal{F}$ in this collection.

翻译：已知对于任意$k\geq 2$，在CONGEST模型中，存在通用蒙特卡洛算法可在$n^{1-1/\Theta(k^2)}$轮内判定$C_{2k}$-无环性。对于$2\leq k\leq 5$，基于对需转发消息数量进行上界估计并在该数量超过特定阈值时中止搜索子程序的方法，已存在运行时间为$O(n^{1-1/k})$轮的更快速蒙特卡洛算法。我们研究了此类基于阈值的算法向更大环检测问题的可能扩展。首先证明，对任意$k\geq 6$，存在包含$2k$环的无限图族，使得任何基于阈值的算法都无法检测该环。特别地，$C_{12}$-无环性与$C_{14}$-无环性均无法通过基于阈值的算法判定。然而，我们进一步证明，$\{C_{12},C_{14}\}$-无环性仍可由基于阈值的算法在$O(n^{1-1/7})= O(n^{0.857\dots})$轮内判定，该运行时间优于通用算法所需的$O(n^{1-1/22})\simeq O(n^{0.954\dots})$轮。此外，我们展示了无限个环族构成的集合，使得该集合中每个环族$\mathcal{F}$的$\mathcal{F}$-无环性均可由基于阈值的算法判定。