Finding Small Complete Subgraphs Efficiently

TL;DR

This paper introduces a simple, efficient algorithm for finding small complete subgraphs in graphs, improving theoretical bounds and practical performance over previous methods, especially for high-arboricity graphs.

Contribution

The paper presents a new simple combinatorial algorithm for triangle listing, extends it to small complete subgraphs, and improves arboricity-sensitive algorithms for counting and detection.

Findings

New simple triangle listing algorithm faster in practice.

Asymptotically tight bounds for listing all copies of K_ell.

Faster algorithms for small K_ell detection in high-arboricity graphs.

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 $O(m \alpha) = O(m^{3/2})$ time, where $\alpha= \alpha(G)$ 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 $\alpha$ in the running time $O(\alpha^{\ell-2} \cdot m)$ 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 $\ell \geq 7$.