Deterministic Even-Cycle Detection in Broadcast CONGEST
Résumé
We show that, for every k ≥ 2, C_{2k}-freeness can be decided in O(n^{1-(1)/(k)}) rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for k = 2 up to logarithmic factors thanks to the lower bound for C₄-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for k ∈ {3,4,5}, and by Fraigniaud et al. [PODC 2024] for k ≥ 6. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without 2k-cycles, which we believe is interesting on its own.