Finding Cycles and Trees in Sublinear Time

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k ≥ 3 and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being Ck-minor free (resp., free from having the corresponding tree-minor). In particular, if the graph is Ω(1)-far from being cycle-free (i.e., a constant fraction of the edges must be deleted to make the graph cycle-free), then the algorithm finds a cycle of polylogarithmic length in time Õ( √ N), where N denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of one-sided error property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of N -vertex graphs can be tested with one-sided error within time complexity Õ(poly(1/ǫ) · √ N), where ǫ denotes the proximity parameter. This matches the known Ω( √ N) query lower bound for one-sided error cycle-freeness testing, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on ǫ. We show that the same upper bound holds for testing whether the input graph has a simple cycle of length at least k, for any k ≥ 3. On the other hand, for any fixed tree T , we show that T -minor freeness has a one-sided error tester of query complexity that only depends on the proximity parameter ǫ. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in o( √ N) complexity.