Apport

Apport is a term that normally refers to the sudden appearance or movement of an item by an unknown force. In NetHack, APPORT is a statistic that represents how likely a pet is to 'fetch' items and bring them to you. Pets with high apport tend to follow you more reliably as well.

Apport can be increased by feeding your pet a treat (corpses do not count) shortly after it has dropped an item near you. You do not necessarily have to throw the treat to the pet, as long as you have handled it recently. For this reason, it is a good practice to pick up tripe in a shop before your carnivorous pet can reach it, then drop it again once your pet has picked something up. When the pet drops the item and then later eats the tripe, you won't be charged for it, and apport should be increased somewhat.

The exact formula for apport gain (calculated when the pet receives a treat) is: 200 / (distance from you to pet when the last item was dropped) + (# of turns since item was dropped)

Ideally, if a pet drops an item right next to you, and you immediately throw it a treat, the calculation would be: 200 / (1 square + 1 turn) = 200/2  = +100 apport

Therefore to maximize apport gain, wait for your pet to drop an item at your feet, then reward it immediately.

Apport is decreased by 1 every time your pet drops an item, so you need to treat your pet from time to time after it drops something, in order to keep apport high. Theoretically, this should be quite easy if you have enough treats, as your pet should be doing a lot more fetching once you have trained it to realize that dropping stuff near you equals a reward.

Knights (who start with a pony and many carrots & apples) or Healers (who can turn rocks in to meatballs) should be able to train their pets to fetch quite well.