geoprocessing - PostGIS: Assign ID of point in layer A to closest point in Layer B

This should have been an obvious precursor (that I didn't ask) to my other question: How to create spider diagrams (hub lines) in PostGIS?

If I do not know the relationship between a point in layer A (stores) and a point in layer B (customers), I would like to generally say "Customer 1 is serviced by the nearest store." While I realize this fact may not be true, it can be a decent surrogate.

Using PostGIS, what is the most efficient way to assign the ID of the nearest point in layer A (stores) to each point in layer B (customers). The output I am looking for is something like below.

Customer | Store
    1    |   A
    2    |   A
    3    |   B
    4    |   C

Answer

likewise:

select A.ID as CUST_ID, (select B.ID from B order by st_distance(A.geom,B.geom) limit 1) as STORE_ID from A