In my experience, it does relieve swelling. Here is a copy/paste from an epsom salts site linked aat the bottom of page.
To reduce the swelling of sprains and bruises or just to enjoy an especially relaxing bath, add two cups of Epsom salts to a tub of warm water. Even better, also add several drops of lavender or neroli oil.
I get good results using one cup, although I see some other sites call for two cups.
Hope this helps.
Corey