A Randomized Algorithm for Long Directed Cycle

Given a directed graph G and a parameter k, the Long Directed Cycle (LDC) problem asks whether G contains a simple cycle on at least k vertices, while the k-Path problems asks whether G contains a simple path on exactly k vertices. Given a deterministic (randomized) algorithm for k-Path as a black box, which runs in time t(G, k), we prove that LDC can be solved in deterministic time O(max{t(G, 2k), 4}) (randomized time O(max{t(G, 2k), 4})). In particular, we get that LDC can be solved in randomized time O(4).