Let $G, G'$ be two symmetric (and positive definite) $n \times n$ real matrices, thought as being some Gram matrices.
Is there an efficient algorithm to decide whether there exists $U \in \mathrm{GL}_n(\Bbb Z)$ such that $G' = {}^{t} U \cdot G \cdot U$ ?
Checking whether two matrices are similar can be done using Jordan normal form I guess, but for congruence (which I thought was called 'equivalence') I don't know. Apparently LLL reduction is not sufficient.