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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.\n  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. 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