Strict control over, as in We told them to keep a tight rein on spending for the next year. This expression alludes to the narrow strap (rein) attached to a bit and used to control a horse's movements. Rein has been used to refer to any kind of restraint since the first half of the 1400s.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.