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 x1, x2, x3 such that x1+x2=2x3. 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. In fact, we consider a more general problem called 3-LDT parameterized by integer parameters α1, α2, α3 and t. In this problem, we need to check if a given set of integers contains distinct elements x1, x2, x3 such that α1 x1+α2 x2 +α3 x3 = t. 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.