🤖 AI Summary
This paper addresses the efficient enumeration of small cliques—particularly triangles and $K_ell$ for $ell geq 3$—in sparse graphs. Methodologically, it introduces a minimalist triangle enumeration algorithm derived from first principles, achieving the $O(malpha)$ time bound, where $alpha$ denotes the graph’s arboricity; it also provides the first rigorous proof that the Chiba–Nishizeki algorithm attains a tight time lower bound for $K_ell$ enumeration, and establishes improved arboricity-sensitive counting and detection bounds for $ell geq 4$. Crucially, the approach abandons reliance on the Nash–Williams theorem, instead reconstructing a combinatorial framework grounded directly in arboricity properties. Theoretically, the results are optimal and asymptotically tight. Experimentally, the triangle enumerator significantly outperforms baselines, while for $K_ell$ with $ell geq 7$, it achieves consistent speedups on 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 $ell geq 7$.