Equivalences between Non-trivial Variants of 3LDT and Conv3LDT
Abstract
The popular 3SUM conjecture states that there is no strongly subquadratic time algorithm for checking if a given set of integers contains three distinct elements x1, x2, x3 such that x1+x2=x3. A closely related problem is to check if a given set of integers contains distinct elements satisfying x1+x2=2x3. This can be reduced to 3SUM in almost-linear time, but surprisingly a reverse reduction establishing 3SUM 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 3LDT 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. We prove that all non-trivial variants of 3LDT over the same universe [-nc,nc] for some c≥2 are equivalent under subquadratic reductions. The main technical tool used in our proof is an application of the famous Behrend's construction that partitions a given set of integers into few subsets that avoid a chosen linear equation. We extend our results to Conv3LDT and show that for all c≥2, all non-trivial variants of 3LDT over the universe [-nc,nc] and of Conv3LDT over the universe [-nc-1,nc-1] are subquadratic-equivalent, so in particular they are all equivalent to 3SUM under subquadratic reductions. Finally, we show how to apply the methods of Fischer et al. to show that we can reduce non-trivial variant of 3LDT (Conv3LDT) over an arbitrary universe to the same variant over cubic (quadratic) universe.