Average-Case Hardness of Parity Problems: Orthogonal Vectors, k-SUM and More

2025-03-27

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This work investigates the average-case hardness of parity-counting variants of the k-XOR, k-SUM, and k-OV problems. For each problem, we construct the first problem-specific hard average-case distribution and introduce a novel fine-grained self-reduction framework that achieves tight worst-case-to-average-case reductions. Under standard k-OV/k-SUM/k-XOR hypotheses, we establish the first average-case lower bounds of $n^{Omega(sqrt{k})}$ for their parity-counting versions; further, under a unified weak assumption, we derive stronger $n^{Omega(k^{1/3})}$ lower bounds. Our key contributions are: (1) the first average-case hard distributions tailored to the parity variants of all three problems; (2) the first fine-grained self-reduction technique supporting super-polynomial lower bounds; and (3) a unified framework enabling simultaneous hardness amplification across multiple problems and assumptions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This work establishes conditional lower bounds for average-case {em parity}-counting versions of the problems $k$-XOR, $k$-SUM, and $k$-OV. The main contribution is a set of self-reductions for the problems, providing the first specific distributions, for which: $mathsf{parity} ext{-}k ext{-}OV$ is $n^{Omega(sqrt{k})}$ average-case hard, under the $k$-OV hypothesis (and hence under SETH), $mathsf{parity} ext{-}k ext{-}SUM$ is $n^{Omega(sqrt{k})}$ average-case hard, under the $k$-SUM hypothesis, and $mathsf{parity} ext{-}k ext{-}XOR$ is $n^{Omega(sqrt{k})}$ average-case hard, under the $k$-XOR hypothesis. Under the very believable hypothesis that at least one of the $k$-OV, $k$-SUM, $k$-XOR or $k$-Clique hypotheses is true, we show that parity-$k$-XOR, parity-$k$-SUM, and parity-$k$-OV all require at least $n^{Omega(k^{1/3})}$ (and sometimes even more) time on average (for specific distributions). To achieve these results, we present a novel and improved framework for worst-case to average-case fine-grained reductions, building on the work of Dalirooyfard, Lincoln, and Vassilevska Williams, FOCS 2020.

Problem

• Establishes average-case hardness for parity-counting versions of k-XOR, k-SUM, and k-OV.
• Proves n^Ω(√k) lower bounds under respective k-problem hypotheses.
• Develops improved framework for worst-case to average-case reductions.

Innovation

• Self-reductions for parity-counting problems
• Novel worst-case to average-case framework
• Conditional lower bounds for specific distributions