  We introduce a simple, accurate, and extremely efficient method for
numerically solving the multi-marginal optimal transport (MMOT) problems
arising in density functional theory. The method relies on (i) the sparsity of
optimal plans [for $N$ marginals discretized by $\ell$ gridpoints each, general
Kantorovich plans require $\ell^N$ gridpoints but the support of optimizers is
of size $O(\ell\cdot N)$ [FV18]], (ii) the method of column generation (CG)
from discrete optimization which to our knowledge has not hitherto been used in
MMOT, and (iii) ideas from machine learning. The well-known bottleneck in CG
consists in generating new candidate columns efficiently; we prove that in our
context, finding the best new column is an NP-complete problem. To overcome
this bottleneck we use a genetic learning method tailormade for MMOT in which
the dual state within CG plays the role of an "adversary", in loose similarity
to Wasserstein GANs. On a sequence of benchmark problems with up to 120
gridpoints and up to 30 marginals, our method always found the exact
optimizers. Moreover, empirically the number of computational steps needed to
find them appears to scale only polynomially when both $N$ and $\ell$ are
simultaneously increased (while keeping their ratio fixed to mimic a
thermodynamic limit of the particle system).
