3SUM Is Really Hard: A Real-to-Integer Reduction

2026-10-07

AI Summary

This study addresses the long-standing open problem in fine-grained complexity regarding real-to-integer reductions for 3SUM, a challenge explicitly posed by Chan et al. By integrating the Frank–Tardos optimization theorem with Freiman-type additive combinatorial results, this work achieves the first tight self-reduction for the problem through rigorous theoretical derivation. The primary contribution establishes that if integer 3SUM admits a subquadratic-time algorithm, then its real-valued counterpart does as well. This result resolves the aforementioned open question and yields the first tight reduction from the real to the integer variant of 3SUM. Consequently, it provides new theoretical bounds and a simplified pathway for future algorithm design concerning 3SUM, advancing the foundational understanding of fine-grained computational complexity.

Abstract

We show that if the 3SUM problem on integer-valued inputs can be solved in truly subquadratic time, then it can also be solved in truly subquadratic time on real-valued inputs. This answers an open problem posed by Chan, Vassilevska Williams, and Xu [STOC 2022], and constitutes the first such tight real-to-integer self-reduction in fine-grained complexity. Our proof relies on a surprising combination of the Frank--Tardos theorem from optimization with Freiman-type theorems from additive combinatorics.

Problem

• 3SUM problem
• fine-grained complexity
• real-to-integer reduction
• subquadratic time

Innovation

• 3SUM problem
• fine-grained complexity
• real-to-integer reduction
• Frank-Tardos theorem
• additive combinatorics