My washer currently drains into my utility sink, which has an 1.25" drain. Based on where it goes into the wall, the sink then seems to drain into the main 3-4" drain for the house along with the bathrooms/kitchen and basement floor drain.
I've found that since getting a new washer a few months ago, I've started to see overflows of the sink because it can't drain fast enough/lint clogging the drain/etc. I've tried using lint traps on the end of the hose to try to at least block the lint, but they fill so quickly that it's not really a feasible long-term solution.
I'm wondering if there's anything I can do to add some sort of better drainage/standpipe for the washer. I know that code would require a 2" or larger standpipe dedicated for the washer, but I'm not sure where that would/should get tied in.
Whereas a longer standpipe (2") might help keep things from overflowing onto the floor, it won't solve the downstream problems that the lint definately creates. If possible, drop a dedicated 2" drain down and tie in downstream of the kitchen, etc.
I had such problems with the exact same thing that you're facing that I finally started draining my washer into the laundry room slop sink and used a large sheet of plastic mesh window screen as a lint filter. That was a rigged fix, but it solved the issue until I got a washer that has a good lint filter. Before that, the drain was full of lint that was the consistency of mashed potatoes. The snake would go right through it but it wouldn't flow downstream. I started using Roebic main line bacterial cleaner and it has worked great for me.