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To apport something is to bring it. 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 had it in your inventory previously. 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 eggs won't splat), and apport should be increased somewhat. Most pets start with an apport of 10.
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:
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. However, it may be helpful to reward your pet immediately once it drops an item anywhere, to raise initial apport high enough for the pet to start seeking you out.
The verb "apport" typically does not appear in dictionaries of the English language. The French language has a verb "apporter"; this word becomes "to apport" in English, and is synonym for "to bring".
In a sentence: My dog apports blessed and uncursed objects.
Because "apport" appears so rarely in English, one may interpret its meaning as more specific than "bring". For example, Wikipedia has an entry for "apport", but only for when a ghost does the apporting!
In the source code of NetHack, apport is the name of a variable field in struct edog. Thus we use "apport" as a noun to refer to the integer in that field. A pet with high apport, performs better to apport objects to you.
In many languages, "apport" is a widely used command in dog training, equivalent to "fetch".