On the Power of Threshold-Based Algorithms for Detecting Cycles in the CONGEST Model

It is known that, for every $k\geq 2$, $C{2k}$-freeness can be decided by a generic Monte-Carlo algorithm running in $n^{{1-1/\Theta(k2)}$} 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.