Equivalences between Non-trivial Variants of 3LDT and Conv3LDT

2020-01-05

AI Summary

This paper establishes subquadratic equivalences among variants of the linear 3-term equation problem (3LDT)—determining whether a set of integers contains three distinct elements satisfying α₁x₁ + α₂x₂ + α₃x₃ = t, where the coefficients αᵢ are not all of the same sign—and its convolutional variant (Conv3LDT). We prove, for the first time, that all nontrivial 3LDT variants are subquadratically equivalent over polynomially bounded universes [−nᶜ, nᶜ] with c ≥ 2, and further show their equivalence to 3SUM. We extend this equivalence to Conv3LDT over the reduced universe [−n^{c−1}, n^{c−1}]. Our approach combines Behrend’s construction, subquadratic reductions, and universe compression techniques. This resolves an open problem posed by Erickson and enables efficient reductions from arbitrary integer universes to cubic or quadratic scales.

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 $x_1, x_2, x_3$ such that $x_1+x_2=x_3$. A closely related problem is to check if a given set of integers contains distinct elements satisfying $x_1+x_2=2x_3$. 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 $alpha_1, alpha_2, alpha_3$ 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 $alpha_1 x_1+alpha_2 x_2 +alpha_3 x_3 = t$. We prove that all non-trivial variants of 3LDT over the same universe $[-n^c,n^c]$ for some $cgeq2$ 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 $cgeq2$, all non-trivial variants of 3LDT over the universe $[-n^c,n^c]$ and of Conv3LDT over the universe $[-n^{c-1},n^{c-1}]$ 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.

Problem

• Establishing equivalence between 3SUM and 3LDT variants
• Proving subquadratic reductions for non-trivial 3LDT problems
• Extending equivalence results to Conv3LDT variants

Innovation

• Reduction establishing 3SUM hardness for linear equations
• Behrend construction partitions sets avoiding equations
• Equivalence proofs for 3LDT variants using subquadratic reductions