Alexander the Great “untied” the knot simply by cutting through it with a sword.
A “Gordian Knot” is any extremely complex problem, and “cutting the Gordian Knot” refers to solving such a problem in a quick decisive manner.
Example: Each year the school counselors face the Gordian knot of scheduling classes.
Example: The councilwoman cut the Gordian knot so that two worthy programs would not have to be cut. Cutting both programs budgets prevented from having to eliminate one or the other.