a 'minimum security prison' [=a prison in which prisoners are allowed more freedom than in most other prisons because they are not considered dangerous or likely to escape]
Ten years is the 'minimum' needed. = We'll need 10 years 'at a/the minimum'. [=we'll need at least 10 years]
We need to keep expenses 'to a (bare) minimum'. [=we need to keep expenses as low as possible]
(..) - Take time. - No, it won't. They play at home tomorrow night. They'll be eight-point favorites, minimum. Now, if he wants, he can win by seven, but no more. But you get him for tomorrow. Can't do that. (..)1974 The Gambler