All non-trivial variants of 3-LDT are equivalent

The popular 3-SUM conjecture states that there is no strongly subquadratic time algorithm for checking if a given set of integers contains three distinct elements that sum up to zero. A closely related problem is to check if a given set of integers contains distinct x 1 , x 2 , x 3 such that x 1 + x 2 = 2x 3 . This can be reduced to 3-SUM in almost-linear time, but surprisingly a reverse reduction establishing 3-SUM hardness was not known.

We provide such a reduction, thus resolving an open question of Erickson [23]. In fact, we consider a more general problem called 3-LDT parameterized by integer parameters α 1 , α 2 , α 3 and t. In this problem, we need to check if a given set of integers contains distinct elements

For some combinations of the parameters, every instance of this problem is a NO-instance or there exists a simple almost-linear time algorithm. We call such variants trivial. We prove that all non-trivial variants of 3-LDT are equivalent under subquadratic reductions. Our main technical contribution is an efficient deterministic procedure based on the famous Behrend's construction that partitions a given set of integers into few subsets that avoid a chosen linear equation.