Min-Plus Convolution Lower Bounds via a Higher-Order BSG Theorem

2026-10-07

AI Summary

This study addresses the long-standing open problem of establishing the equivalence of Min-Plus convolution assumptions in fine-grained complexity and deriving conditional lower bounds for multiple classical problems. To this end, the work proposes a universe reduction technique alongside a higher-order BSG theorem to extract strong additive structure from weakly structured sets. By integrating additive combinatorics with refined APSP-equivalence techniques, it overcomes fundamental barriers inherent to arithmetic settings. The primary contributions include proving the equivalence of two convolution assumptions under standard hypotheses, thereby yielding tight conditional lower bounds for problems such as Min-Max convolution. Furthermore, this research develops a truly subquadratic-time algorithm for low-rank 3SUM that unifies all previously known special cases.

Abstract

Min-Plus Convolution is a central problem in fine-grained complexity, and the associated Min-Plus Convolution Hypothesis forms the basis for a wide range of conditional lower bounds for fundamental problems. It is closely connected to the APSP and 3SUM Hypotheses, and in fact implies both, making it a unifying hypothesis for two of the main pillars of the area. In this work we establish several strong results related to Min-Plus Convolution. We design a universe reduction, showing, under a plausible additive combinatorics assumption, that the Min-Plus Convolution Hypothesis is equivalent to the Strong Min-Plus Convolution Hypothesis. We also obtain tight conditional lower bounds for multiple long-standing problems, including Min-Max Convolution and Bounded Monotone Min-Plus Convolution. Our approach is inspired by Fischer's recent equivalence between several variants of APSP [STOC '26], but extending that technique to the arithmetic setting requires overcoming deep obstacles. To this end, we develop a novel additive structure theorem that can be viewed as a higher-order substitute of the Balog-Szemerédi-Gowers (BSG) theorem, allowing us to extract strong additive structure even from weakly structured sets. Building on this structural result, we show that certain structured 3SUM instances (namely, sets with low rank) can be solved in truly subquadratic time. This algorithm forms the main algorithmic ingredient in our reductions. Besides, it generalizes all previously known truly subquadratic-time special cases of 3SUM, and is therefore of independent interest.

Problem

• Min-Plus Convolution
• Fine-grained complexity
• Conditional lower bounds
• 3SUM
• Additive combinatorics

Innovation

• Min-Plus Convolution
• Higher-Order BSG Theorem
• Additive Structure Theorem
• Fine-Grained Complexity
• 3SUM