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Fast Matrix Multiplication meets the Submodular Width

Mahmoud Abo-Khamis, Xiao Hu, Dan Suciu

TL;DR

The paper addresses the data-complexity of evaluating Boolean conjunctive queries and introduces the omega-submodular width, a parameter that seamlessly integrates fast matrix multiplication (MM) with combinatorial techniques. By modeling MM complexity via edge-dominated polymatroids and extending variable elimination orders to generalized variable elimination orders (GVEOs), the authors define a unified, information-theoretic width that captures both TD-based planning and MM-based acceleration. They show how to compute this width through a finite set of linear programs and prove that any query can be answered in time tilde O(N^{omega-subw(Q)}), recovering best-known results for several query classes while also yielding new, faster algorithms for some classes. The framework thus provides a principled, general approach to leveraging MM in database query processing and paves the way for extensions to counting/summing queries and broader semiring settings. Overall, the work offers a rigorous bridge between combinatorial join theory and fast linear-algebraic techniques, with meaningful implications for database optimizers and theoretical bounds.

Abstract

One fundamental question in database theory is the following: Given a Boolean conjunctive query Q, what is the best complexity for computing the answer to Q in terms of the input database size N? When restricted to the class of combinatorial algorithms, it is known that the best known complexity for any query Q is captured by the submodular width of Q. However, beyond combinatorial algorithms, certain queries are known to admit faster algorithms that often involve a clever combination of fast matrix multiplication and data partitioning. Nevertheless, there is no systematic way to derive and analyze the complexity of such algorithms for arbitrary queries Q. In this work, we introduce a general framework that captures the best complexity for answering any Boolean conjunctive query Q using matrix multiplication. Our framework unifies both combinatorial and non-combinatorial techniques under the umbrella of information theory. It generalizes the notion of submodular width to a new stronger notion called the omega-submodular width that naturally incorporates the power of fast matrix multiplication. We describe a matching algorithm that computes the answer to any query Q in time corresponding to the omega-submodular width of Q. We show that our framework recovers the best known complexities for Boolean queries that have been studied in the literature, to the best of our knowledge, and also discovers new algorithms for some classes of queries that improve upon the best known complexities.

Fast Matrix Multiplication meets the Submodular Width

TL;DR

The paper addresses the data-complexity of evaluating Boolean conjunctive queries and introduces the omega-submodular width, a parameter that seamlessly integrates fast matrix multiplication (MM) with combinatorial techniques. By modeling MM complexity via edge-dominated polymatroids and extending variable elimination orders to generalized variable elimination orders (GVEOs), the authors define a unified, information-theoretic width that captures both TD-based planning and MM-based acceleration. They show how to compute this width through a finite set of linear programs and prove that any query can be answered in time tilde O(N^{omega-subw(Q)}), recovering best-known results for several query classes while also yielding new, faster algorithms for some classes. The framework thus provides a principled, general approach to leveraging MM in database query processing and paves the way for extensions to counting/summing queries and broader semiring settings. Overall, the work offers a rigorous bridge between combinatorial join theory and fast linear-algebraic techniques, with meaningful implications for database optimizers and theoretical bounds.

Abstract

One fundamental question in database theory is the following: Given a Boolean conjunctive query Q, what is the best complexity for computing the answer to Q in terms of the input database size N? When restricted to the class of combinatorial algorithms, it is known that the best known complexity for any query Q is captured by the submodular width of Q. However, beyond combinatorial algorithms, certain queries are known to admit faster algorithms that often involve a clever combination of fast matrix multiplication and data partitioning. Nevertheless, there is no systematic way to derive and analyze the complexity of such algorithms for arbitrary queries Q. In this work, we introduce a general framework that captures the best complexity for answering any Boolean conjunctive query Q using matrix multiplication. Our framework unifies both combinatorial and non-combinatorial techniques under the umbrella of information theory. It generalizes the notion of submodular width to a new stronger notion called the omega-submodular width that naturally incorporates the power of fast matrix multiplication. We describe a matching algorithm that computes the answer to any query Q in time corresponding to the omega-submodular width of Q. We show that our framework recovers the best known complexities for Boolean queries that have been studied in the literature, to the best of our knowledge, and also discovers new algorithms for some classes of queries that improve upon the best known complexities.

Paper Structure

This paper contains 42 sections, 31 theorems, 89 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorial Join Algorithms
  4. Beyond Combinatorial Join Algorithms
  5. Our Contributions
  6. Paper Outline
  7. Overview
  8. Q1: Translating MM complexity into polymatroids
  9. Q2: Query plans for MM
  10. Defining the $\omega$-submodular width
  11. Computing the $\omega$-submodular width
  12. Computing query answers in $\omega$-subdmodular width time
  13. Preliminaries
  14. The $\omega$-Submodular Width: Formal Definition
  15. Generalizing variable elimination orders
  16. ...and 27 more sections

Key Result

proposition 1

Given $\mathcal{H} = (\mathcal{V}, \mathcal{E})$ where $k \stackrel{\mathrm{def}}{=} |\mathcal{V}|$:

Figures (4)

  • Figure 1: The proof sequence for the Shannon inequality \ref{['eq:intro:shannon:triangle']} along with the corresponding algorithm for $Q_\triangle$.
  • Figure 2: Diagram of the polymatroid for the clique-3 hypergraph.
  • Figure 3: Diagram of the polymatroid for cycle-4 hypergraph.
  • Figure 4: Diagram of the polymatroid for 3-Pyramid hypergraph.

Theorems & Definitions (64)

  • proposition 1: Equivalence of TDs and VEOs DBLP:conf/pods/KhamisNR16
  • definition 1: Generalized Variable Elimination Order (GVEO)
  • definition 2: Matrix multiplication expression, $\mathsf{MM}$
  • proposition 2
  • proof
  • proposition 3
  • proof
  • definition 3: Variable elimination expression via matrix multiplication, $\mathsf{EMM}$
  • definition 4: $\omega$-submodular width
  • proposition 4
  • ...and 54 more