We consider the problem of approximating the girth, $g$, of an unweighted and undirected graph $G=(V,E)$ with $n$ nodes and $m$ edges. A seminal result of Itai and Rodeh [SICOMP'78] gave an additive $1$-approximation in $O(n^2)$ time, and the main open question is thus how well we can do in subquadratic time. In this paper we present two main results. The first is a $(1+\varepsilon,O(1))$-approximation in truly subquadratic time. Specifically, for any $k\ge 2$ our algorithm returns a cycle of length $2\lceil g/2\rceil+2\left\lceil\frac{g}{2(k-1)}\right\rceil$ in $\tilde{O}(n^{2-1/k})$ time. This generalizes the results of Lingas and Lundell [IPL'09] who showed it for the special case of $k=2$ and Roditty and Vassilevska Williams [SODA'12] who showed it for $k=3$. Our second result is to present an $O(1)$-approximation running in $O(n^{1+\varepsilon})$ time for any $\varepsilon > 0$. Prior to this work the fastest constant-factor approximation was the $\tilde{O}(n^{3/2})$ time $8/3$-approximation of Lingas and Lundell [IPL'09] using the algorithm corresponding to the special case $k=2$ of our first result.

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