Which treatments for breast abscess are most cost-effective?

Antibiotics. Antibiotics with warm compresses are most cost effective, but an abscess, almost by definition, is difficult for antibiotics to penetrate and kill the bacteria. Therefore, you're almost always going to have to have the abscess surgically treated. That may mean draining it with a needle, but often requires surgically cutting out the abscess wall.
Depends. It would be necessary to know the cause of the breast abscess and its sensitivity to antibiotics and the potential need for surgical drainage to answer this question fully.