There is a hidden stash of booze in the house somewhere or you have booze out in the open where he is 'nipping' into. He is not relapsing. He is an alcoholic.
If you don't drink (I hope not) get all the alcohol out of the house. It is time to stop dreaming and start taking action for yourself and your children's well being.
You are not in a good situation unless he sees what damage he is doing with his drinking...the lies, the false, promises, the denial is distructive. You need to have a plan and the right tools to make yourself powerful against his addiction. Seek support.