How does dragging a body affect your AP?
Solution 1:
By default it costs 3 times as much as normal movement. The strength upgrades, and the augment can all substract 0.5 from that, meaning that if you have all 3, it costs 1.5 times as much as normal movement.
The game tracks fractional AP, and displays it rounded down. Normal movement costs 1 AP for adjacent, 1.41 AP (square root 2) for diagonal square.
Source: I looked in the source code, and it was also discussed on the forums.