Finding Even Cycles Faster Via Capped K-Walks

Finding cycles in graphs is a fundamental problem in algorithmic graph theory. In this paper, we consider the problem of finding and reporting a cycle of length 2k in an undirected graph G with n nodes and m edges for constant k >= 2. A classic result by Bondy and Simonovits [J. Combinatorial Theory, 1974] implies that if m >= 100kn(1+1/k), then G contains a 2k-cycle, further implying that one needs to consider only graphs with m = O(n(1+1/k)).Previously the best known algorithms were an O(n(2)) algorithm due to Yuster and Zwick [J. Discrete Math 1997] as well as a O(m(2)-((1+[k/2]-1)/(k+1))) algorithm by Alon et. al. [Algorithmica 1997].We present an algorithm that uses O(m(2k/(k+1))) time and finds a 2k-cycle if one exists. This bound is O(n(2)) exactly when m = Theta(n(1+1/k)). When finding 4-cycles our new bound coincides with Alon et. al., while for every k > 2 our new bound yields a polynomial improvement in m.Yuster and Zwick noted that it is "plausible to conjecture that O(n(2)) is the best possible bound in terms of n". We show "conditional optimality": if this hypothesis holds then our O(m(2k/(k+1))) algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a 6-cycle in time O(m(3/2-epsilon)) for any epsilon > 0 unless boolean matrix multiplication can be solved combinatorially in time O(n(3-epsilon')) for some epsilon' > 0, which is widely believed to be false. Coupled with our main result, this gives tight bounds for finding 6-cycles combinatorially and also separates the complexity of finding 4- and 6-cycles giving evidence that the exponent of m in the running time should indeed increase with k.The key ingredient in our algorithm is a new notion of capped k-walks, which are walks of length k that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.