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Fast Approximate Counting of Cycles

Keren Censor-Hillel, Tomer Even, Virginia Vassilevska Williams

TL;DR

The paper presents a fast, approximate counting framework for fixed-length cycles in directed graphs, achieving a running time of $ ilde{O}( extsf{MM}(n, n/t^{1/(h-2)}, n))$ to estimate the number of $h$-cycles for any constant $h≥3$ and fixed accuracy. It introduces a simplified, path-like Recursive Template that relies on two black boxes: Find-Heavy and Count-Heavy, enabling a robust single-path recursion and a doubling variant to adapt to unknown counts. The heavy-vertex identification leverages nonuniform sampling and rectangular matrix multiplication, while counting heavy copies uses color-coding and careful avoidance of double counting to attain a $(1±ε)$-approximation with additive polylog factors. A tight conditional lower bound based on fine-grained hypotheses shows the proposed running times are essentially optimal for both triangles and general $h$-cycles, with extensions to directed and odd cycles. Overall, the work advances both triangle counting and general $h$-cycle counting, offering nearly optimal, t-dependent runtimes and practical strategies for sparse graphs via matrix-multiplication-based techniques.

Abstract

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a $(1 \pm \eps)$-approximation in $\tilde{O}(n^ω/t^{ω-2})$ time, where $t$ is the unknown number of triangles in the given $n$ node graph and $ω<2.372$ is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an $n\times n/t$ matrix by an $n/t \times n$ matrix. We then extend our framework to obtain the first nontrivial $(1 \pm \eps)$-approximation algorithms for the number of $h$-cycles in a graph, for any constant $h\geq 3$. Our running time is \[\tilde{O}(\mathsf{MM}(n,n/t^{1/(h-2)},n)), \textrm{the time to multiply } n\times \frac{n}{t^{1/(h-2)}} \textrm{ by } \frac{n}{t^{1/(h-2)}}\times n \textrm{ matrices}.\] Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

Fast Approximate Counting of Cycles

TL;DR

The paper presents a fast, approximate counting framework for fixed-length cycles in directed graphs, achieving a running time of to estimate the number of -cycles for any constant and fixed accuracy. It introduces a simplified, path-like Recursive Template that relies on two black boxes: Find-Heavy and Count-Heavy, enabling a robust single-path recursion and a doubling variant to adapt to unknown counts. The heavy-vertex identification leverages nonuniform sampling and rectangular matrix multiplication, while counting heavy copies uses color-coding and careful avoidance of double counting to attain a -approximation with additive polylog factors. A tight conditional lower bound based on fine-grained hypotheses shows the proposed running times are essentially optimal for both triangles and general -cycles, with extensions to directed and odd cycles. Overall, the work advances both triangle counting and general -cycle counting, offering nearly optimal, t-dependent runtimes and practical strategies for sparse graphs via matrix-multiplication-based techniques.

Abstract

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a -approximation in time, where is the unknown number of triangles in the given node graph and is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an matrix by an matrix. We then extend our framework to obtain the first nontrivial -approximation algorithms for the number of -cycles in a graph, for any constant . Our running time is Finally, we show that under popular fine-grained hypotheses, this running time is optimal.
Paper Structure (24 sections, 29 theorems, 11 equations, 2 figures, 2 algorithms)

This paper contains 24 sections, 29 theorems, 11 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Technical Overview
  4. I. The Recursive Template.
  5. II. Finding the Heavy Vertices.
  6. III. Counting the Heavy Copies.
  7. Roadmap:
  8. Preliminaries
  9. The Recursive Template
  10. Organization.
  11. The $\mathsf{Template}$ Algorithm.
  12. The $\mathsf{Doubling\mhyphen Template}$ Algorithm.
  13. Proof of \ref{['lemma:correctness wrap']} using \ref{['lemma:simp']}
  14. Proving \ref{['lemma:simp']} Using a More Refined Version
  15. Proof of \ref{['lemma2:ih']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $G$ be a given graph with $n$ vertices and let $h\geq 3$ be a fixed integer. There is a randomized algorithm that outputs an approximation $\hat{t}$ for the number $t$ of $h$-cycles in $G$ such that ${\mathrm{Pr}}\left[ (1-\varepsilon)t \leq \hat{t}\leq (1+\varepsilon)t \right]\geq 1-1/n^2$, for

Figures (2)

  • Figure 1: A comparison between our new running times for approximate triangle counting with prior work, together with the lower bounds, both conditional and unconditional.
  • Figure 2: Comparative Runtime Analysis

Theorems & Definitions (56)

  • Theorem 1.1: Approximating the Number of $h$-Cycles
  • Theorem 1.2
  • Definition 1: Fast Matrix Multiplication Definitions
  • Theorem 1.3: Chernoff Bound dubhashi2009concentration
  • Theorem 1.4: Chebyshev's Inequality
  • Lemma 2.0: Guarantees for the $\algCoreNP$ Algorithm
  • Remark 1
  • Lemma 2.0: Guarantees for the $\algWrapNP$ Algorithm
  • Claim 1: The event $\mathcal{E}_{\mathrm{Find\mhyphen Heavy}}$
  • Claim 2
  • ...and 46 more