  We study algorithms called rank-revealers that reveal a matrix's rank
structure. Such algorithms form a fundamental component in matrix compression,
singular value estimation, and column subset selection problems. While
column-pivoted QR has been widely adopted due to its practicality, it is not
always a rank-revealer. Conversely, Gaussian elimination (GE) with a pivoting
strategy known as global maximum volume pivoting is guaranteed to estimate a
matrix's singular values but its exponential complexity limits its interest to
theory. We show that the concept of local maximum volume pivoting is a crucial
and practical pivoting strategy for rank-revealers based on GE and QR. In
particular, we prove that it is both necessary and sufficient; highlighting
that all local solutions are nearly as good as the global one. This insight
elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer,
and we implement a version that is observed to be at most $2\times$ more
computationally expensive than CPQR. We unify the landscape of rank-revealers
by considering GE and QR together and prove that the success of any pivoting
strategy can be assessed by benchmarking it against a local maximum volume
pivot.
