Finding Small Complete Subgraphs Efficiently

(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(mα) = 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. (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^ℓ-2· m) of the algorithm of Chiba and Nishizeki for listing all copies of K_ℓ, where ℓ≥ 3, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of K_ℓ, for small ℓ≥ 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 ℓ≥ 7.