I think you negotiate with them and conclude that with their many talents, they can work extra hard, get a raise, and pay for a cleaning lady/guy to come once a week.  Of course, how expensive this is will depend a lot on where you live. If you're not working (e.g. you're both still in college), this may seem extravagant. But if there is really no other issue (this is not masking some problem about you feeling your partner takes you for granted, etc...) why not take the simple way out? 

Next time you are seething, ask yourself "If my partner suddenly decided to drop everything and clean the house top to bottom, would I drop $30 to take us both out to dinner and celebrate?" If the answer is yes consider the money available and call the cleaning service. Depending on your personality this scenario might go the other way: "Imagine right now I stopped nagging you about cleaning..."

I once knew someone quite wealthy who said that the nice thing about being rich was that your time was completely your own. Now, I'm not saying we live in a society where the landed elite devote themselves to philosophy and the betterment of mankind, but I've thought about this remark many times since. Most people (I include myself) spend an astonishing amount of time greasing the gears of their lives so that everything else will run. 

If most couples I know had a slightly larger apartment and slightly more delegation of chores to other people, it seems to me that 90% of their arguments would be avoided. Perhaps the fights would just move elsewhere, but I'm not so sure. 

I might get labeled as a red-blooded free marketeer for this last remark. Don't get me wrong, I'm not saying there isn't another solution, only that the other solution takes a long and winding road through psychology, social conditioning, deep-seated reform of relation between the sexes etc and Amanda might be asking about the short route. 

I think this is a major problem in relationships. Not just cleaning, of course (though this is particularly annoying, and if it's a man being messy and a woman cleaning up, it opens all sorts of old wounds about gender roles and servitude and is almost impossible to deal with rationally). But what about the more general issue that very often you love someone with whom it's quite difficult to live? Perhaps they like to get up in the morning and listen to the news while you need quiet to sleep in, or maybe they have to have the window open while you prefer the bedroom completely dark. Or you need space and light while your partner is not in the least bothered by clutter, and seems to be a clutter magnet. There are all sorts of ways in which two people can be fundamentally incompatible when it comes to living in the same space. And how to resolve it? Living apart isn't really the answer. There's no substitute for waking up with the person you love. But at the same time it's important not to forget how important basic needs like quiet, space (and music, freedom to make noise) are, and how much it alters your life when these are removed.

As a practical matter, the only solution most people I know have found is time: you both get older, your jobs pay better, you can afford a bit more space, or you have kids and all bets are off about those quiet mornings in any case.

I'm not really sure this was an answer; but it's an interesting, if difficult, problem.

If my choice were to live in a pig-stye  with my partner, I would opt for 2 separate living spaces.