We study the Online Connectivity Augmentation Problem (OCAP) with cut arrivals. In this problem, we are given a $k$-edge-connected graph $G = (V,E)$ with $|V| = n$ and a set of links $L \subseteq \binom{V}{2}$. In each iteration, a minimum cut arrives to be augmented, and we seek to maintain a smallest set of links crossing all such cuts. This problem is a special case of online hitting set in which the ground set is $L$ and a sequence of sets to be hit arrive from $\mathcal{C} = \{δ_L(C) : |δ_E(C)| = k\}$. As such, OCAP admits a $O(\log^2 n)$-competitive algorithm using the classical algorithm for Online Set Cover of Alon, Awerbuch, Azar, Buchbinder, and Naor. If $k$ is odd, then OCAP reduces to the case where $G$ is a tree. This case was studied by Naor, Umboh, and Williamson, who gave an optimal $O(\log n)$-competitive algorithm, which even holds in weighted setting. Recently, Garg and Subramanian gave an $O(\log n)$-competitive algorithm for unweighted OCAP which holds in the popular node-arrival model, but not for cut-arrivals. No better than $O(\log^2 n)$-competitive algorithm is known for weighted OCAP. In this paper, we provide a nearly optimal $O(\log n \log \log n)$-competitive algorithm for weighted OCAP under cut-arrivals and an optimal $O(\log n)$-competitive algorithm for unweighted OCAP with cut-arrivals. We also give an $O(1)$ algorithm for Online Cycle Augmentation under node-arrivals. Since no $o(\log n)$-competitive algorithm is possible on a cycle with cut-arrivals, this establishes a separation between the best competitive ratio achievable in the two arrival models.

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