Girlfriend cheated on me again but she has an alcohol problem?

I'm not married but I have a live-in girlfriend of 4 years. I took this week off from work to visit friends in Philly. The thing is I left my laptop and some work papers behind so I stopped at Trenton and went back to New York to get it. It was really late at night and I ended up walking in on my girlfriend and another guy. She was very drunk and we argued a bit but I ended up leaving for philly anyway. The thing is she's an alcoholic and she's done this before- she goes out to a bar and when she's drunk she loses her morals and ends up letting a guy come home with her. I'm a bit torn on what to do when I return home tomorrow. On one side I love her and she has a alcohol addiction but on the other I want a woman who is with me and only me. Do you think I should ask her to go to AA counseling and forgive her and see what happens or do you think her AA problem is irrelevant and I shouldn't put up with a cheating gf?