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Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection

Bartłomiej Dudek, Nick Fischer, Geri Gokaj, Ce Jin, Marvin Künnemann, Xiao Mao, Mirza Redžić

Abstract

We revisit the complexity of verifying basic identities, such as associativity and distributivity, on a given finite algebraic structure. In particular, while Rajagopalan and Schulman (FOCS'96, SICOMP'00) gave a surprising randomized algorithm to verify associativity of an operation $\odot: S\times S\to S$ in optimal time $O(|S|^2)$, they left the open problem of finding any subcubic algorithm for verifying distributivity of given operations $\odot,\oplus: S\times S\to S$. Our results are as follows: * We resolve the open problem by Rajagopalan and Schulman by devising an algorithm verifying distributivity in strongly subcubic time $O(|S|^ω)$, together with a matching conditional lower bound based on the Triangle Detection Hypothesis. * We propose arithmetic progression detection in small universes as a consequential algorithmic challenge: We show that unless we can detect $4$-term arithmetic progressions in a set $X\subseteq\{1,\dots, N\}$ in time $O(N^{2-ε})$, then (a) the 3-uniform 4-hyperclique hypothesis is true, and (b) verifying certain identities requires running time~$|S|^{3-o(1)}$. * A careful combination of our algorithmic and hardness ideas allows us to \emph{fully classify} a natural subclass of identities: Specifically, any 3-variable identity over binary operations in which no side is a subexpression of the other is either: (1) verifiable in randomized time $O(|S|^2)$, (2) verifiable in randomized time $O(|S|^ω)$ with a matching lower bound from triangle detection, or (3) trivially verifiable in time $O(|S|^3)$ with a matching lower bound from hardness of 4-term arithmetic progression detection. * We obtain near-optimal algorithms for verifying whether a given algebraic structure forms a field or ring, and show that \emph{counting} the number of distributive triples is conditionally harder than verifying distributivity.

Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection

Abstract

We revisit the complexity of verifying basic identities, such as associativity and distributivity, on a given finite algebraic structure. In particular, while Rajagopalan and Schulman (FOCS'96, SICOMP'00) gave a surprising randomized algorithm to verify associativity of an operation in optimal time , they left the open problem of finding any subcubic algorithm for verifying distributivity of given operations . Our results are as follows: * We resolve the open problem by Rajagopalan and Schulman by devising an algorithm verifying distributivity in strongly subcubic time , together with a matching conditional lower bound based on the Triangle Detection Hypothesis. * We propose arithmetic progression detection in small universes as a consequential algorithmic challenge: We show that unless we can detect -term arithmetic progressions in a set in time , then (a) the 3-uniform 4-hyperclique hypothesis is true, and (b) verifying certain identities requires running time~. * A careful combination of our algorithmic and hardness ideas allows us to \emph{fully classify} a natural subclass of identities: Specifically, any 3-variable identity over binary operations in which no side is a subexpression of the other is either: (1) verifiable in randomized time , (2) verifiable in randomized time with a matching lower bound from triangle detection, or (3) trivially verifiable in time with a matching lower bound from hardness of 4-term arithmetic progression detection. * We obtain near-optimal algorithms for verifying whether a given algebraic structure forms a field or ring, and show that \emph{counting} the number of distributive triples is conditionally harder than verifying distributivity.

Paper Structure

This paper contains 58 sections, 61 theorems, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Conditionally Optimal Verification of Distributivity
  4. A Classification Theorem via 4-Term Arithmetic Progressions
  5. Arithmetic Progression Detection as a Fine-Grained Hypothesis
  6. Main Classification Theorem
  7. Further Results on Ring and Field Verification and Counting
  8. Counting Variant
  9. Conclusion and Open Problems
  10. Preliminaries
  11. Algebraic Expressions
  12. Subcubic Complexity of Distributivity Checking
  13. Distributivity Checking is Triangle-Hard
  14. Distributivity Checking via Triangle Counting
  15. Technical Overview: Beyond Distributivity Checking
  16. ...and 43 more sections

Key Result

theorem 1.1

Given a set $S$ and two binary operations $\oplus, \odot: S\times S\to S$, there is a randomized $\mathcal{O}(n^{\omega})$-time algorithm that decides with high probability whether for all $a,b,c\in S$, we have $a\odot (b\oplus c) = (a \odot b) \oplus (a\odot c)$. Any $O(n^{\omega-\epsilon})$-time a

Figures (2)

  • Figure 1: Tree representation of the expression $f(a,b,c) \equiv (a \odot b) \oplus (c \odot a)$, with ${\rm depth}(f) = 2$ and ${\rm var}(f) = \{a,b,c\}$.
  • Figure 2: Illustrates the known fine-grained reductions between $k$-AP Detection and other central hard problems.

Theorems & Definitions (133)

  • theorem 1.1: Distributivity Verification
  • theorem 1.2
  • theorem 1.3: Classification Theorem, informal version
  • theorem 1.4
  • theorem 1.5
  • theorem 1.6
  • theorem 3.1
  • lemma 3.2
  • proof
  • claim 1
  • ...and 123 more