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

Mina Dalirrooyfard, Andrea Lincoln, Barna Saha, Virginia Vassilevska Williams

TL;DR

This work establishes conditional average-case lower bounds for parity-counting versions of core fine-grained problems (k-SUM, k-OV, k-XOR) via self-reductions and a simplified worst-case to average-case framework. By moving from modular polynomial representations to integer polynomials and introducing fine $d$-degree polynomials, the authors create a pathway from worst-case hardness under rETH/SETH/k-XOR/k-SUM/k-OV/k-Clique to average-case hardness on natural, easy-to-sample distributions. They prove $N^{\Omega(\sqrt{K})}$-hardness for parity-$K$-OV/XOR/SUM under respective hypotheses, and $N^{\Omega(K^{1/3})}$-hardness under broader collective hypotheses, with explicit reductions between factored and unfactored variants. The results advance understanding of the average-case difficulty of central fine-grained problems and provide a framework potentially extensible to additional problems and distributions. Overall, the paper offers the first substantial average-case lower bounds for parity versions of these foundational problems and shows the power of a streamlined factored-to-unfactored reduction strategy in fine-grained complexity.

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: $\mathsf{parity}\text{-}k\text{-}OV$ is $n^{Ω(\sqrt{k})}$ average-case hard, under the $k$-OV hypothesis (and hence under SETH), $\mathsf{parity}\text{-}k\text{-}SUM$ is $n^{Ω(\sqrt{k})}$ average-case hard, under the $k$-SUM hypothesis, and $\mathsf{parity}\text{-}k\text{-}XOR$ is $n^{Ω(\sqrt{k})}$ 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 $n^{Ω(k^{1/3})}$ (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.

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

TL;DR

This work establishes conditional average-case lower bounds for parity-counting versions of core fine-grained problems (k-SUM, k-OV, k-XOR) via self-reductions and a simplified worst-case to average-case framework. By moving from modular polynomial representations to integer polynomials and introducing fine -degree polynomials, the authors create a pathway from worst-case hardness under rETH/SETH/k-XOR/k-SUM/k-OV/k-Clique to average-case hardness on natural, easy-to-sample distributions. They prove -hardness for parity--OV/XOR/SUM under respective hypotheses, and -hardness under broader collective hypotheses, with explicit reductions between factored and unfactored variants. The results advance understanding of the average-case difficulty of central fine-grained problems and provide a framework potentially extensible to additional problems and distributions. Overall, the paper offers the first substantial average-case lower bounds for parity versions of these foundational problems and shows the power of a streamlined factored-to-unfactored reduction strategy in fine-grained complexity.

Abstract

This work establishes conditional lower bounds for average-case {\em parity}-counting versions of the problems -XOR, -SUM, and -OV. The main contribution is a set of self-reductions for the problems, providing the first specific distributions, for which: is average-case hard, under the -OV hypothesis (and hence under SETH), is average-case hard, under the -SUM hypothesis, and is average-case hard, under the -XOR hypothesis. Under the very believable hypothesis that at least one of the -OV, -SUM, -XOR or -Clique hypotheses is true, we show that parity--XOR, parity--SUM, and parity--OV all require at least (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.

Paper Structure

This paper contains 25 sections, 17 theorems, 8 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Overview
  4. Comparison to Prior Work
  5. Preliminaries
  6. Factored Problems
  7. Factored Vector factoredProblems
  8. Factored Problems factoredProblems
  9. Un-factored Problems
  10. Framework Definitions
  11. Framework
  12. Reductions Between Factored Problems
  13. More efficient reductions between factored problems
  14. Even More Efficient Reductions to k-SUM
  15. Worst-Case to Average-Case Fine-Grained Problems
  16. ...and 10 more sections

Key Result

Theorem 1.1

Assuming that at least one of the worst-case $k$-OV, $k$-SUM, or $k$-XOR hypotheses holds, all of $\mathsf{parity}\text{-}k\text{-}OV$, $\mathsf{parity}\text{-}k\text{-}SUM$ and $\mathsf{parity}\text{-}k\text{-}XOR$ are $n^{\Omega(\sqrt{k})}$-hard on average for a natural distribution.

Figures (5)

  • Figure 1: An example of a factored $k$-OV instance, where each factored vector has $g$ sets of $4$-bit numbers ($b=4$), see $\vec{v}$ in list $i$ as an example for a factored vector. In reducing a factored $k$-OV instance $I$ to a $kg$-OV instance $I^*$, we take each set in factored vectors as a new list, as specified by the red markers.
  • Figure 2: Roadmap of the approach to prove average-case complexity for the $K$-$Q$ problem from $k$-$P$ problem. Each problem is $k$ (or $K$) partite, and the size of the problems refer to the number of vectors or factored vectors in each partition.
  • Figure 3: Reductions to average case $K$-OV of size $N$. The size of the unfactored problems is mentioned as a parameter in front of them. To see what the values of $b,g,n$ and $k$ are in terms of $K$ and $N$ see Table \ref{['tab:ov-reduction-param-values']}.
  • Figure 4: Reductions to average case $\oplus K$-XOR of size $N$. The size of the unfactored problems is mentioned as a parameter in front of them. To see what the values of $b,g,n$ and $k$ are in terms of $K$ and $N$ see Table \ref{['tab:xor-reduction-param-values']}.
  • Figure 5: Reductions to average case $K$-SUM of size $N$. The size of the unfactored problems is mentioned as a parameter in front of them. To see what the values of $b,g,n$ and $k$ are in terms of $K$ and $N$ see Table \ref{['tab:sum-reduction-param-values']}.

Theorems & Definitions (27)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Definition 2.1: The $k$-clique Hypothesis
  • Definition 2.2: The $k$-XOR Hypothesis
  • Definition 2.3: The $k$-SUM Hypothesis C3sum
  • Definition 2.4: (Strong) Exponential Time Hypothesiscseth
  • Definition 2.5: The $k$-OV Hypothesis virgiSurvey
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 17 more