Of the major chain home stores, which place is cheapest for carpet installation?

I would go to an actual carpet store to have it done. The big box stores that you mentioned have reputations of only hiring hacks. This is because they keep a large portion of what they charge you for installation for themselves and hire the cheapest contractors they can find to do the install. So why do you think these contractors would do work for these big box stores for a fraction of what they could make working for anyone else...because NO ONE else would hire them. The cost difference is so minimal if there is even a difference that is not worth the chance.

Home Depot

I've had carpet installed many times and from many places, the best jobs that were done were from ads in the local newspaper that were from retired carpet layers needing a few extra dollars and enjoying what they were doing.