  We study the multi-marginal partial optimal transport (POT) problem between
$m$ discrete (unbalanced) measures with at most $n$ supports. We first prove
that we can obtain two equivalence forms of the multimarginal POT problem in
terms of the multimarginal optimal transport problem via novel extensions of
cost tensor. The first equivalence form is derived under the assumptions that
the total masses of each measure are sufficiently close while the second
equivalence form does not require any conditions on these masses but at the
price of more sophisticated extended cost tensor. Our proof techniques for
obtaining these equivalence forms rely on novel procedures of moving mass in
graph theory to push transportation plan into appropriate regions. Finally,
based on the equivalence forms, we develop optimization algorithm, named
ApproxMPOT algorithm, that builds upon the Sinkhorn algorithm for solving the
entropic regularized multimarginal optimal transport. We demonstrate that the
ApproxMPOT algorithm can approximate the optimal value of multimarginal POT
problem with a computational complexity upper bound of the order
$\tilde{\mathcal{O}}(m^3(n+1)^{m}/ \varepsilon^2)$ where $\varepsilon > 0$
stands for the desired tolerance.
