What algorithms are known to perform the task of updating a database by inserting, updating, and deleting rows in the presence of database constraints?
More specifically
Why are you even trying to do this? The correct way to do it is to get the database engine to defer the checking of the constraints until the transaction is committed.
The problem that you pose is intractable in the general case. If you consider just a transitive closure of the foreign keys in the rows you want to update in database then it is only possible to solve this where the graph describes a tree. If there is a cycle in the graph and you can break the cycle by replacing a foreign key value with a NULL then you can re-write one SQL and add another to later update the column. If you can't replace a key value with a NULL then it can't be solved.
As I say, the correct way to do this is to turn off the constraints until all of the SQL has been run and then turn them back on for the commit. The commit will fail if the constraints aren't met. Postgres (for example) has a feature which makes this very easy.