  Beautiful formulas are known for the expected cost of random two-dimensional
assignment problems, but in higher dimensions even the scaling is not known. In
three dimensions and above, the problem has natural "Axial" and "Planar"
versions, both of which are NP-hard. For 3-dimensional Axial random assignment
instances of size $n$, the cost scales as $\Omega(1/n)$, and a main result of
the present paper is a linear-time algorithm that, with high probability, finds
a solution of cost $O(n^{-1+o(1)})$. For 3-dimensional Planar assignment, the
lower bound is $\Omega(n)$, and we give a new efficient matching-based
algorithm that with high probability returns a solution with cost $O(n \log
n)$.
