Is zakat obligatory on a flock of sheep that is fed at the owner's expense for half the year or less, knowing that zakat is due on sheep that graze?

Zakat is not obligatory on livestock that are fed for half the year or more. This is the preponderant opinion. However, if the feeding is for less than that, then zakat becomes obligatory on them. The Malikis hold that feeding does not waive zakat at all.