On the power of threshold-based algorithms for detecting cycles in the CONGEST model

It is known that, for every k >= 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 <= k <= 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 >= 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...)) rounds, which is faster than using the generic algorithm, which would run in O(n(1) (1/122)) similar or equal to O(n(0.954...)) rounds. Moreover, we exhibit an infinite collection of families of cycles such that threshold-based algorithms can decide F-freeness for every F in this collection.