  We analyze two algorithms for approximating the general optimal transport
(OT) distance between two discrete distributions of size $n$, up to accuracy
$\varepsilon$. For the first algorithm, which is based on the celebrated
Sinkhorn's algorithm, we prove the complexity bound
$\widetilde{O}\left({n^2/\varepsilon^2}\right)$ arithmetic operations. For the
second one, which is based on our novel Adaptive Primal-Dual Accelerated
Gradient Descent (APDAGD) algorithm, we prove the complexity bound
$\widetilde{O}\left(\min\left\{n^{9/4}/\varepsilon, n^{2}/\varepsilon^2
\right\}\right)$ arithmetic operations. Both bounds have better dependence on
$\varepsilon$ than the state-of-the-art result given by
$\widetilde{O}\left({n^2/\varepsilon^3}\right)$. Our second algorithm not only
has better dependence on $\varepsilon$ in the complexity bound, but also is not
specific to entropic regularization and can solve the OT problem with different
regularizers.
