  In 1900, as a part of his 18th problem, Hilbert proposed the question to
determine the densest congruent (or translative) packings of a given solid,
such as the unit ball or the regular tetrahedron of unit edges. Up to now, our
knowledge about this problem is still very limited, excepting the ball case. It
is conjectured that, for some particular solids such as tetrahedra,
cuboctahedra and octahedra, their maximal translative packing densities and
their maximal lattice packing densities are identical. To attack this
conjecture, this paper suggests a computer approach to determine the maximal
local translative packing density of a given polytope, by studying associated
color graphs and applying optimization. In particular, all the tetrahedral
case, the cuboctahedral case and the octahedral case of the conjecture have
been reduced into finite numbers of manageable optimization problems.
