In the definition of 3-PARTITION of Garey&Johnson, the instance is a set of $3m$ integers such that the sum of all these integers is $mB$ and such that each integer is strictly between $B/4$ and $B/2$. This problem is strongly NP-hard.
The special case DISTINCT 3-PARTITION when all the integers are distinct has been studied here (at the end of the paper). The problem is also NP-hard in the strong sense. But in the definition of the instance, the authors do not include the condition that all the integers must strictly be between $B/4$ and $B/2$.
Is it clear that even with this condition DISTINCT 3-PARTITION is still strongly NP-hard? Thank you!
