Finding Small Complete Subgraphs Efficiently

TL;DR

This work advances the study of listing and detecting small complete subgraphs by presenting a simple Hybrid algorithm that lists all triangles in $O(m α)=O(m^{3/2})$ time, where $α$ is the graph arboricity. It extends the approach to enumerate copies of $K_ℓ$ and proves the $O(α^{ℓ-2} m)$ bound is tight in terms of $m$ and $α$, while delivering improved arboricity-sensitive running times for counting/detecting small $K_ℓ$ (notably for $ℓ≥4$ and high arboricity). The paper also provides constructions showing tightness, and experimental results indicating practical speedups over previous methods. Overall, it offers both simple, implementable algorithms and a unified framework combining extension, triangle, and edge-count methods to yield improved performance across a broad range of graphs and subgraph sizes.

Abstract

(I) We revisit the algorithmic problem of finding all triangles in a graph $G=(V,E)$ with $n$ vertices and $m$ edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in $O(m α) = O(m^{3/2})$ time, where $α= α(G)$ is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. Our experimental results show that our simple algorithm for triangle listing is substantially faster in practice than that of Chiba and Nishizeki on all examples of real-world graphs we tried. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on $m$ and $α$ in the running time $O(α^{\ell-2} \cdot m)$ of the algorithm of Chiba and Nishizeki for listing all copies of $K_\ell$, where $\ell \geq 3$, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of $K_\ell$, for small $\ell \geq 4$. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every $\ell \geq 7$.

Figures (1)

• Figure 1: Illustration of the graph $G$ for $k=3$, $b=4$.

Theorems & Definitions (8)

• Theorem 1
• Corollary 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof