How to check if two date ranges overlap in mysql?

If we are guaranteed that date_started, datefinished, $DateA and $DateB are not NULL, and we're guaranteed that date_started is not greater than date_finished...

`s` represents `date_started`
`f` represents `date_finished`
`a` represents the smaller of `$DateA` and `$DateB`
`b` represents the larger of `$DateA` and `$DateB`

Visually:

      s-----f       overlap
 -----+-----+-----  -------  
  a-b |     |        NO
  a---b     |        YES
  a-----b   |        YES
  a---------b        YES
  a-----------b      YES
      a---b |        YES
      a-----b        YES
      a-------b      YES
      | a-b |        YES
      | a---b        YES     
      | a-----b      YES     
      |     a-b      YES
      |     | a-b    NO

We can easily detect when there's no "overlap" of the ranges:

( a > f OR b < s )

And we can easily negate that to return "true" when there is an "overlap":

NOT ( a > f OR b < s )

Converting that to SQL:

NOT ( GREATEST('{$dateA}','{$dateB}') < p.date_started
      OR LEAST('{$dateA}','{$dateB}') > p.date_finished
    )