It depends. First, cedar chips take a very long time to decompose, and will affect the soil and water, but not significantly differently than the decomposition of cedar in the forests which occurs naturally. However, I suggest that you avoid trying to decompose them with other materials such as garden waste or household vegetable matter.
I assume by "Household Compost" you are talking about vegetable/fruit waste from inside the home. In this case, the important thing is to ensure that the composting is done in a container that is fully protected from rats in particular. Some things, like red worms, significantly improve the efficiency of composting these wastes.
It is best that you have good drainage to ensure the composted material dries out fairly well. It is a good idea to turn it once in a while, i.e. mix it up. There is a neat tool which makes this quit easy. Done properly, this will not adversly affect the water or soil.
The one thing that I would NOT include in my compost pile is evergreen clippings, needles or cones. These don't compost very well, and will make the soil too acidic.