Equivalences between Non-trivial Variants of 3LDT and Conv3LDT

TL;DR

The paper resolves open questions about the hardness landscape of 3LDT and Conv3LDT by proving that all non-trivial variants over polynomial-size universes are subquadratic-equivalent and, in particular, subquadratic-equivalent to 3SUM. Central to the approach is Behrend-style progression-free partitioning, which enables controlled reductions that separate trivial cases and handle zero-sum instances, together with color-coding to simulate distinct-element selections. The authors extend these reductions across Conv3LDT variants and bridge 3LDT and Conv3LDT, including universe-size reductions to cubic and quadratic scales, thereby tying a broad family of linear-degeneracy problems to the 3SUM conjecture. The results yield a cohesive framework showing that Average is subquadratic-equivalent to 3SUM and that Conv3LDT variants form a tightly connected equivalence class under subquadratic reductions, with practical implications for hardness reductions in related convolution and array-structured problems.

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 $α_1, α_2, α_3$ 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 $α_1 x_1+α_2 x_2 +α_3 x_3 = t$. We prove that all non-trivial variants of 3LDT over the same universe $[-n^c,n^c]$ for some $c\geq2$ 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 $c\geq2$, 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.

Figures (3)

• Figure 1: Subquadratic reductions between different variants of 3LDT. Subscripts to coefficients $\alpha$ and $t$ denote if the reduction allows changing the coefficients or not. In all the reductions the value of parameter $c$ is preserved and hence not shown.
• Figure 2: Subquadratic reductions between different variants of Conv3LDT. Subscripts to coefficients $\alpha$ and $t$ denote if the reduction allows changing the coefficients or not. All the reductions preserve the parameter $c\geq 1$ describing the size of the universe.
• Figure 3: Subquadratic reductions between different variants of Conv3LDT and 3LDT. Subscripts to coefficients $\alpha$ and $t$ denote if the reduction allows changing the coefficients or not. Only two reductions change the value of parameter $c$, all other reductions preserve it.

Theorems & Definitions (66)

• Definition 3: cf. WilliamsW18
• Definition 4
• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Proposition 5
• proof