In the online survivable network design problem, a graph with edge costs is known in advance, and requests arrive one at a time. Each request is a pair of vertices $(s,t)$ with a connectivity requirement $k$, and the algorithm must irrevocably buy edges so that $s$ and $t$ are joined by $k$ edge-disjoint paths. The goal is to keep the total cost close to that of an optimal offline solution. Gupta, Krishnaswamy, and Ravi~\cite{GKR12} obtain an $O(k\log^3 n)$-competitive algorithm, where $k$ is the maximum requirement, without relaxing the connectivity requirement. Dehghani, Ehsani, Hajiaghayi, Liaghat, and Seddighin~\cite{DEHLS18} obtain a polylogarithmic competitive ratio when the connectivity requirement is allowed to be relaxed by a factor of $2+\varepsilon$. We show that relaxing the connectivity requirement by a factor of only $1+\varepsilon$ suffices to obtain a polylogarithmic competitive ratio.

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