通过高阶 BSG 定理的 Min-Plus 卷积下界
Min-Plus Convolution Lower Bounds via a Higher-Order BSG Theorem
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中文总结 AI 辅助
本文通过高阶 BSG 定理证明 Min-Plus 卷积假设与强假设等价，并推导多个下界，同时提出解决低秩 3SUM 的次二次算法。
中文摘要 AI 辅助
Min-Plus 卷积是细粒度复杂性的核心问题，相关的 Min-Plus 卷积假设构成了许多基本问题条件下界的基础。它与 APSP 和 3SUM 假设紧密相关，实际上蕴含这两者，使其成为该领域两大支柱的统一假设。在这项工作中，我们建立了与 Min-Plus 卷积相关的若干强结果。我们设计了一个论域归约，在合理的加性组合学假设下，证明了 Min-Plus 卷积假设等价于强 Min-Plus 卷积假设。我们还为多个长期存在的问题（包括 Min-Max 卷积和有界单调 Min-Plus 卷积）获得了紧的条件下界。我们的方法受到 Fischer 最近关于 APSP 若干变体等价性的研究 [STOC '26] 的启发，但将该技术扩展到算术设置需要克服深层障碍。为此，我们发展了一个新颖的加性结构定理，可视为 Balog-Szemerédi-Gowers (BSG) 定理的高阶替代，使我们能够从弱结构集合中提取强加性结构。基于这一结构结果，我们证明了某些结构化的 3SUM 实例（即低秩集合）可以在真正次二次时间内求解。该算法是我们归约中的主要算法成分。此外，它推广了所有先前已知的 3SUM 真正次二次时间特例，因此具有独立的意义。
英文摘要
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.
发表机构
- Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
- University of California, Berkeley（加州大学伯克利分校）
- University of California, San Diego（加州大学圣地亚哥分校）
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