Subquadratic time approximation algorithms for the girth

We study the problem of determining the girth of an unweighted undirected graph. We obtain several new efficient approximation algorithms for graphs with n nodes and m edges and unknown girth g. We consider additive and multiplicative approximations. Additive Approximations. We present: • an Õ(n ³/m)-time algorithm which returns a cycle of length at most g + 2 if g is even and g + 3 if g is odd. This complements the seminal work of Itai and Rodeh [SIAM J. Computing'78] who gave an algorithm that in O(n²) time finds a cycle of length g if g is even, and g + 1 if g is odd. • an Õ(n³/m)-time algorithm which returns a cycle of length at most g′ + 2, where g′ is the length of the shortest even cycle in G. This result complements the work of Yuster and Zwick [SIAM J. Discrete Math'97] who showed how to compute g′ in O(n²) time. Multiplicative Approximations. We present: • an Õ(n^5/3)-time algorithm which returns a cycle of length at most 3g/2 + z/2 when g is even and 3g/2 + z/2 + 1 when g is odd, where z = -g mod 4, z ∈ {0,1,2,3}. This gives an Õ(n^5/3)-time 2-approximation for the girth, the first subquadratic 2-approximation algorithm, resolving an open question of Lingas and Lundell [IPL'09]. • an O(n^1.968)-time (8/5)-approximation algorithm for the girth in graphs with girth at least 4 (i.e., triangle-free graphs). This is the first subquadratic time (2 - ε)-approximation algorithm for the girth for triangle-free graphs, for any ε > 0. We prove that a deterministic algorithm of this kind is not possible for directed graphs, thus showing a strong separation between undirected and directed graphs for girth approximation.