We give various deterministic polynomial time reductions among
approximation problems on point lattices. Our reductions are
both efficient and robust, in the sense that they preserve the
rank of the lattice and approximation factor achieved. Our main
result shows that for any *g >= 1*, approximating
*all* the *successive minima* of a lattice (and,
in particular, approximately solving the *Shortest
Independent Vectors Problem*, SIVP_{g}) within a
factor *g* reduces under deterministic polynomial time
rank-preserving reductions to approximating the *Closest
Vector Problem* (CVP) within the same factor *g*.
This solves an open problem posed by Blomer in (ICALP
2000). As an application, we obtain faster algorithms
for the exact solution of SIVP that run in time *n!
s ^{O(1)}* (where