Last summer and fall I fought this same problem myself. Between my own 3 (2 Boxers, 1 mixed breed), a foster here and there and sometimes my 2 Boxer grandbabies, my yard turned into a disaster area. That's not to mention all those feet to clean and the constant carpet cleaning indoors. I tried resodding twice. It looked great for a short while but within a couple of weeks, it was back to being a dirt/mud pit. When the fall rains hit, I was ready to scream. So, when the rain settled down and the ground dried out some, I tilled up the yard to relieve some of the soil compaction and just mulched in the whole area with a hardwood mulch. I put the mulch down at probably about 4" or so over the area. It has made such a difference! No more mud, it really doesn't look bad at all and even when it rains, the dogs still don't have that much on their paws to clean off. I sometimes get a little of the mulch on the floor indoors from their paws, but it's very little and it vacuums or sweeps up easily. I'm going to make an attempt to start some ground cover in the hopes of making it look more attractive with some color. There are some ground covers that can withstand the abuse. I'll only be able to do a little at a time as to do it all at once would be cost prohibitive for me given the size of the area.
A friend of mine with 3 Boxers was having the same issue in her small city yard. She already had a small brick patio and just last week she finished extending the patio to cover most of the yard, leaving some areas for flower beds. I haven't seen it yet, but so far she's quite happy with the results.
