I worked for a major carpet manufacturer for two years straight out of college.
Your best bet is tile. If you don't want to do that and you do want to keep the dogs, you need to go with a olefin berber carpet.
Olefin is the material, and it is the cheapest of the four major materials that are used to make carpet. Olefin is also known as polypropylene.
Other four are PET (polyester - often from recycled coke bottles), Nylon, and wool.
Olefin, PET, and Nylon are all made out of oil/plastic. As such, the worst thing you can do is clean them with something like Resolve that is oil based. Oil+oil = base for dirt to stick. So you clean with resolve and then it is bigger and dirtier than ever before - that's the oil in resolve providing a place for dirt to stick.
Olefin is the same material that waterski ropes are made out of. Is is not as soft or springy as nylon or PET, however the big benefits of it are how you can clean it and the price. Olefin carpet is usually used to make cheap looped carpet. Examples are the short pile looped carpet in a church or office building OR the residential berber looped carpet. Something like this:
or this:
Generally, the bigger the loop, the softer it will feel. But it will never feel as soft as a think frieze.
Olefin is solution dyed, which means they melt the plastic olefin down, add the dyes and then solidify it, creating a yarn. They are very thin strings, but each one actually runs miles long. Because they dying process is done when it is a liquid, each yarn will not change colors. This means you can clean it with anything, and that includes bleach. Yes, you can bleach these carpets with pure bleach and it won't affect it negatively in anyway. You cannot do this with nylon, PET or wool.
So get a cheap olefin berber that is acceptable, get a waterproof pad under it, and bleach it when they have an accident. There are nylon and PET berbers, so make sure it is olefin (polypropylene.)
