The Shortest Lattice Vector (SLV) problem is in general hard to solve, except
for special cases (such as root lattices and lattices for which an obtuse
superbase is known). In this paper, we present a new class of SLV problems that
can be solved efficiently. Specifically, if for an n-dimensional lattice, a
Gram matrix is known that can be written as the difference of a diagonal matrix
and a positive semidefinite matrix of rank k (for some constant k), we show
that the SLV problem can be reduced to a k-dimensional optimization problem
with countably many candidate points. Moreover, we show that the number of
candidate points is bounded by a polynomial function of the ratio of the
smallest diagonal element and the smallest eigenvalue of the Gram matrix.
Hence, as long as this ratio is upper bounded by a polynomial function of n,
the corresponding SLV problem can be solved in polynomial complexity. Our
investigations are motivated by the emergence of such lattices in the field of
Network Information Theory. Further applications may exist in other areas.Comment: 13 page