All non-trivial variants of 3-LDT are equivalent
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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. 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 x_1, x_2, x_3 such that α_1 x_1+α_2 x_2 +α_3 x_3 = 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.
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