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

Abstract

It is known that, for every $$k\ge 2$$ , $$C_{2k}$$ -freeness can be decided by a generic Monte-Carlo algorithm running in $$n^{1-1/\varTheta (k^2)}$$ rounds in the congest model. For $$2\le k\le 5$$ , 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 $$k\ge 6$$ , 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.