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We study structured convex optimization problems, with additive objective$r:=p + q$, where $r$ is ($\mu$-strongly) convex, $q$ is $L_q$-smooth andconvex, and $p$ is $L_p$-smooth, possibly nonconvex. For such a class ofproblems, we proposed an inexact accelerated gradient sliding method that canskip the gradient computation for one of these components while still achievingoptimal complexity of gradient calls of $p$ and $q$, that is,  $\mathcal{O}(\sqrt{L_p/\mu})$ and $\mathcal{O}(\sqrt{L_q/\mu})$,respectively. This result is much sharper than the classic black-box complexity$\mathcal{O}(\sqrt{(L_p+L_q)/\mu})$, especially when the difference between$L_q$ and $L_q$ is large. We then apply the proposed method to solvedistributed optimization problems over master-worker architectures, underagents' function similarity, due to statistical data similarity or otherwise.The distributed algorithm achieves for the first time lower complexity boundson {\it both} communication and local gradient calls, with the former havingbeing a long-standing open problem. Finally the method is extended todistributed saddle-problems (under function similarity) by means of solving aclass of variational inequalities, achieving lower communication andcomputation complexity bounds.

Optimal Gradient Sliding and its Application to Distributed Optimization Under Similarity