  For a family of probability spaces
$\{(X_k,\mathcal{B}_{X_k},\mu_k)\}_{k=1}^N$ and a cost function $c:
X_1\times\cdots\times X_N\to \mathbb{R}$ we consider the Monge-Kantorovich
problem \begin{align*}\tag{MK}\label{MONKANT}
\inf_{\lambda\in\Pi(\mu_1,\ldots,\mu_N)}\int_{\prod_{k=1}^N X_k}c\,d\lambda.
\end{align*} Then for each ordered subset
$\mathcal{P}=\{i_1,\ldots,i_p\}\subsetneq\{1,...,N\}$ with $p\geq 2$ we create
a new cost function $c_\mathcal{P}$ corresponding to the original cost function
$c$ defined on $\prod_{k=1}^p X_{i_k}$. This new cost function $c_\mathcal{P}$
enjoys many of the features of the original cost $c$ while it has the property
that any optimal plan $\lambda$ of \eqref{MONKANT} restricted to $\prod_{k=1}^p
X_{i_k}$ is also an optimal plan to the problem
\begin{align*}\tag{RMK}\label{REDMONKANT}
\inf_{\tau\in\Pi(\mu_{i_1},\ldots\mu_{i_p})}\int_{\prod_{k=1}^p
X_{i_k}}c_{\mathcal{P}}\,d\tau. \end{align*} Our main contribution in this
paper is to show that, for appropriate choices of index set $\mathcal{P}$, one
can recover the optimal plans of \eqref{MONKANT} from \eqref{REDMONKANT}. In
particular, we study situations in which the problem \eqref{MONKANT} admits a
unique solution depending on the uniqueness of the solution for the lower
marginal problems of the form \eqref{REDMONKANT}. This allows us to prove many
uniqueness results for multi-marginal problems when the unique optimal plan is
not necessarily induced by a map. To this end, we extensively benefit from
disintegration theorems and the $c$-extremality notions. Moreover, by employing
this argument, besides recovering many standard results on the subject
including the pioneering work of Gangbo-\'Swi\c ech, several new applications
will be demonstrated to evince the applicability of this argument.
