We have "heritage crosses"...mixes of a lot of heritage breeds. A few are mostly Tamworth...they do very well on pasture and like to root a lot. We have a Large Black/Hampshire cross boar. We have two Duroc crosses. And children from all of the above. They all do equally well on pasture - the Tamworths are a little more docile and a little smaller, a little more interested in forage than grain. The Durocs are good growers and will happily plump themselves up on grain all day long. Both have benefits, obviously...depends on what you want to do with them. All of these breeds have been fantastic mothers, needing very little help from us in any aspect.
Right now we have them on a 4-acre pasture, eating the native greens, grains and roots. We accent their diets with oats and corn (hog mash for sows when they are close to giving birth or breastfeeding). We will look into seeding the pasture with mangels, turnips, grains, etc. once they finish clearing the land of its goldenrod and scrub.
Also: I have heard that they cannot eat parsnips as they irritates their mouths and noses.
They do get a lot out of the corn; if we are feeding whole kernels maybe 5-10% of it goes undigested and reseeds. (Higher for oats...maybe 30% undigested and reseeded). Usually we feed cracked corn, which they seem to digest more completely.
I use manure that's been sitting for 3-6 months to put around the fruit trees, and it's really mostly hay. I only get the stuff when we've had them indoors and they are always deeply bedded, either because it is winter or because they were having piglets...most of their manure ends up right on the pasture. So I have "used" straw/hay that's also got some goat poop, rabbit poop and chicken poop mixed into it. Then chickens are in charge of turning the compost pile, and they do a pretty good job of it.
They do pick areas where they tend to concentrate their toilet duties, but they spread it about a good bit as well. Mostly they just tend to keep it away from their current favorite sleeping spots and the spot where we feed them grain.
Once we get a stable herd size, we are going to look into getting into the dairy waste stream, as applestar mentioned. Whey from yogurt processing places, "expired" milk from the dairy up the street, etc. That is a very good way to get calories and flavor into pigs. They also adore crabapples and nuts; I am planning to plant a lot of crabapples and black walnut trees etc. throughout the pasture for them next spring.