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Triangle Detection in H-Free Graphs

Amir Abboud, Ron Safier, Nathan Wallheimer

TL;DR

This work investigates combinatorial Triangle Detection in fixed-$H$-free graphs, aiming to beat cubic time by exploiting structural properties of the forbidden pattern. A central embedding framework anchored around a triangle, together with the notion of nice colorings, yields subcubic algorithms for a broad family of embeddable patterns, with runtimes of the form $\tilde{O}(n^{3-1/2^{k-3}})$ (triangle-free) or $\tilde{O}(n^{3-1/2^{k-4}})$ (contains a triangle). The authors establish lower bounds under the BMM conjecture for patterns that are not $3$-colorable or contain multiple triangles, and they extend the results to $H$-sensitive and $C_{2k+1}$-free graphs, including specialized results for $C_5$ and triangle listing. They also connect to generalized Turán-type questions through ex$(K_3,H)$ and discuss the listing problem, showing that the algorithms extend to enumerating all triangles with comparable time when possible. Overall, the paper advances a potential dichotomy for $H$-free Triangle Detection and provides a suite of techniques for structured worst-case instances, with several open questions about extending the classification and derandomization.

Abstract

We initiate the study of combinatorial algorithms for Triangle Detection in $H$-free graphs. The goal is to decide if a graph that forbids a fixed pattern $H$ as a subgraph contains a triangle, using only "combinatorial" methods that notably exclude fast matrix multiplication. Our work aims to classify which patterns admit a subcubic speedup, working towards a dichotomy theorem. On the lower bound side, we show that if $H$ is not $3$-colorable or contains more than one triangle, the complexity of the problem remains unchanged, and no combinatorial speedup is likely possible. On the upper bound side, we develop an embedding approach that results in a strongly subcubic, combinatorial algorithm for a rich class of "embeddable" patterns. Specifically, for an embeddable pattern of size $k$, our algorithm runs in $\tilde O(n^{3-\frac{1}{2^{k-3}}})$ time, where $\tilde O(\cdot)$ hides poly-logarithmic factors. This algorithm also extends to listing all the triangles within the same time bound. We supplement this main result with two generalizations: 1) A generalization to patterns that are embeddable up to a single obstacle that arises from a triangle in the pattern. This completes our classification for small patterns, yielding a dichotomy theorem for all patterns of size up to eight. 2) An $H$-sensitive algorithm for embeddable patterns, which runs faster when the number of copies of $H$ is significantly smaller than the maximum possible $Ω(n^k)$. Finally, we focus on the special case of odd cycles. We present specialized Triangle Detection algorithms that are very efficient: 1) A combinatorial algorithm for $C_{2k+1}$-free graphs that runs in $\tilde O(m+n^{1+2/k})$ time for every $k\geq2$, where $m$ is the number of edges in the graph. 2) A combinatorial $C_5$-sensitive algorithm that runs in $\tilde O(n^2+n^{4/3}t^{1/3})$ time, where $t$ is the number of $5$-cycles in the graph.

Triangle Detection in H-Free Graphs

TL;DR

This work investigates combinatorial Triangle Detection in fixed--free graphs, aiming to beat cubic time by exploiting structural properties of the forbidden pattern. A central embedding framework anchored around a triangle, together with the notion of nice colorings, yields subcubic algorithms for a broad family of embeddable patterns, with runtimes of the form (triangle-free) or (contains a triangle). The authors establish lower bounds under the BMM conjecture for patterns that are not -colorable or contain multiple triangles, and they extend the results to -sensitive and -free graphs, including specialized results for and triangle listing. They also connect to generalized Turán-type questions through ex and discuss the listing problem, showing that the algorithms extend to enumerating all triangles with comparable time when possible. Overall, the paper advances a potential dichotomy for -free Triangle Detection and provides a suite of techniques for structured worst-case instances, with several open questions about extending the classification and derandomization.

Abstract

We initiate the study of combinatorial algorithms for Triangle Detection in -free graphs. The goal is to decide if a graph that forbids a fixed pattern as a subgraph contains a triangle, using only "combinatorial" methods that notably exclude fast matrix multiplication. Our work aims to classify which patterns admit a subcubic speedup, working towards a dichotomy theorem. On the lower bound side, we show that if is not -colorable or contains more than one triangle, the complexity of the problem remains unchanged, and no combinatorial speedup is likely possible. On the upper bound side, we develop an embedding approach that results in a strongly subcubic, combinatorial algorithm for a rich class of "embeddable" patterns. Specifically, for an embeddable pattern of size , our algorithm runs in time, where hides poly-logarithmic factors. This algorithm also extends to listing all the triangles within the same time bound. We supplement this main result with two generalizations: 1) A generalization to patterns that are embeddable up to a single obstacle that arises from a triangle in the pattern. This completes our classification for small patterns, yielding a dichotomy theorem for all patterns of size up to eight. 2) An -sensitive algorithm for embeddable patterns, which runs faster when the number of copies of is significantly smaller than the maximum possible . Finally, we focus on the special case of odd cycles. We present specialized Triangle Detection algorithms that are very efficient: 1) A combinatorial algorithm for -free graphs that runs in time for every , where is the number of edges in the graph. 2) A combinatorial -sensitive algorithm that runs in time, where is the number of -cycles in the graph.

Paper Structure

This paper contains 43 sections, 31 theorems, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results for Triangle Detection in $H$-free graphs
  3. $H$-sensitive algorithms
  4. A special focus on $C_{2k+1}$-free graphs
  5. Generalized Turán problem and Triangle Listing
  6. Paper organization.
  7. Technical Overview
  8. The embedding approach
  9. An algorithm for nicely colored patterns
  10. The case of an attached triangle.
  11. The approach for $H$-sensitive algorithms.
  12. A specialized $C_5$-sensitive algorithm.
  13. The approach for $C_{2k+1}$-free graphs.
  14. Related Works
  15. Complexity dichotomies in $H$-free graphs
  16. ...and 28 more sections

Key Result

theorem 1

If $H$ is not $3$-colorable, or if it contains more than one triangle, then combinatorial Triangle Detection algorithms for $H$-free graphs require $n^{3-o(1)}$ time under conj:bmm.

Figures (3)

  • Figure 1: An $8$-vertex triangle-free graph that is not $C_5$-colorable, but admits a nice coloring (illustrated).
  • Figure 2: The graph from vsurimova2020adynamic, together with a proper $3$-coloring (illustrated). This graph does not admit a nice coloring, and in every proper $3$-coloring of it, the neighborhood of every vertex is two-colored.
  • Figure 3: A minimal example of a pattern which does not fall into any of the categories of \ref{['prop:smallpatterns']}. Namely, the pattern is $3$-colorable (illustrated) and contains exactly one triangle. In every proper $3$-coloring of it, there are no vertices with a monochromatic neighborhood. Moreover, every vertex of the triangle has degree greater than $2$ and therefore it does not fall into the classification of \ref{['thm:attached']}.

Theorems & Definitions (45)

  • Conjecture 1: Combinatorial Triangle Detection (equivalently, the BMM Conjecture)
  • theorem 1: Lower bounds classes
  • proposition 1.1
  • Definition 1.1: Nice colorings
  • theorem 2
  • proposition 1.1
  • theorem 3
  • proposition 1.1
  • theorem 4
  • theorem 5
  • ...and 35 more