A Note on the Conditional Optimality of Chiba and Nishizeki's Algorithms
Yael Kirkpatrick, Surya Mathialagan
TL;DR
This work establishes conditional lower bounds that match the classic Chiba–Nishizeki subgraph listing algorithms. By reductions from the $3SUM$ and Exact-$k$-Clique hypotheses, it shows that 4-cycle listing is tight when arboricity is below a threshold, and that $k$-clique listing is essentially optimal for $α=n^{1-δ}$. The results use reductions from All-Edge Sparse Triangle and employ 4-cycle-free constructions, tripartite color-coding, and edge-weight hashing to transfer hardness to the targeted problems. Overall, the paper explains why no polynomial improvements have emerged for these fundamental subgraph listing tasks and delineates the precise boundary where Chiba–Nishizeki’s algorithms remain optimal in a fine-grained complexity sense.
Abstract
In a seminal work, Chiba and Nishizeki [SIAM J. Comput. `85] developed subgraph listing algorithms for triangles, 4-cycle and $k$-cliques, where $k \geq 3.$ The runtimes of their algorithms are parameterized by the number of edges $m$ and the arboricity $α$ of a graph. The arboricity $α$ of a graph is the minimum number of spanning forests required to cover it. Their work introduces: * A triangle listing algorithm that runs in $O(mα)$ time. * An output-sensitive 4-Cycle-Listing algorithm that lists all 4-cycles in $O(mα+ t)$ time, where $t$ is the number of 4-cycles in the graph. * A k-Clique-Listing algorithm that runs in $O(mα^{k-2})$ time, for $k \geq 4.$ Despite the widespread use of these algorithms in practice, no improvements have been made over them in the past few decades. Therefore, recent work has gone into studying lower bounds for subgraph listing problems. The works of Kopelowitz, Pettie and Porat [SODA `16] and Vassilevska W. and Xu [FOCS `20] showed that the triangle-listing algorithm of Chiba and Nishizeki is optimal under the $\mathsf{3SUM}$ and $\mathsf{APSP}$ hypotheses respectively. However, it remained open whether the remaining algorithms were optimal. In this note, we show that in fact all the above algorithms are optimal under popular hardness conjectures. First, we show that the $\mathsf{4}\text{-}\mathsf{Cycle}\text{-}\mathsf{Listing}$ algorithm is tight under the $\mathsf{3SUM}$ hypothesis following the techniques of Jin and Xu [STOC `23], and Abboud, Bringmann and Fishcher [STOC `23] . Additionally, we show that the $k\text{-}\mathsf{Clique}\text{-}\mathsf{Listing}$ algorithm is essentially tight under the exact $k$-clique hypothesis by following the techniques of Dalirooyfard, Mathialagan, Vassilevska W. and Xu [STOC `24]. These hardness results hold even when the number of 4-cycles or $k$-cliques in the graph is small.