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The sensitivity of optimal transport (OT) to noise has motivated the study of robust variants. In this paper, we study two such formulations of semi-discrete OT in $\mathbb{R}^d$: (i) the $\alpha$-optimal partial transport, which minimizes the cost of transporting a mass of $\alpha$; and (ii) the $\lambda$-robust optimal transport, which regularizes the OT problem using the total variation (TV) distance. First, we provide a novel characterization of the optimal solutions in these settings, showing they can be represented as a restricted Laguerre diagram. Second, we exploit this characterization to establish a strong algorithmic connection between the two problems, showing that any solver for one can be adapted to solve the other with comparable precision. Third, we overcome key challenges posed in extending the cost-scaling paradigm to compute these variants of OT and present an algorithm that computes the exact solution up to $\log (1/\varepsilon)$ bits of precision in $n^{O(d)}\log (1/\varepsilon)$ time, where $n$ is the support size of the discrete distribution. Finally, we present an $n^{1+o(1)}\varepsilon^{-O(d)}$ time approximation algorithm for the above variants of OT.