  Given a matrix $A$ and $k\geq 0$, we study the problem of finding the
$k\times k$ submatrix of $A$ with the maximum determinant in absolute value.
This problem is motivated by the question of computing the determinant-based
lower bound of [LSV86] on hereditary discrepancy, which was later shown to be
an approximate upper bound as well [Mat13]. The special case where $k$
coincides with one of the dimensions of $A$ has been extensively studied.
[Nik15] gave a $2^{O(k)}$-approximation algorithm for this special case,
matching known lower bounds; he also raised as an open problem the question of
designing approximation algorithms for the general case.
  We make progress towards answering this question by giving the first
efficient approximation algorithm for general $k\times k$ subdeterminant
maximization with an approximation ratio that depends only on $k$. Our
algorithm finds a $k^{O(k)}$-approximate solution by performing a simple local
search. Our main technical contribution, enabling the analysis of the
approximation ratio, is an extension of Pl\"ucker relations for the
Grassmannian, which may be of independent interest; Pl\"ucker relations are
quadratic polynomial equations involving the set of $k\times k$ subdeterminants
of a $k\times n$ matrix. We find an extension of these relations to $k\times k$
subdeterminants of general $m\times n$ matrices.
