Listing 4-Cycles

TL;DR

This paper introduces an efficient algorithm for listing all 4-cycles in a graph with a time complexity that surpasses previous bounds for triangle listing, and it is conditionally tight based on recent complexity conjectures.

Contribution

The paper presents a novel algorithm for listing all 4-cycles in a graph with a tight time complexity, separating it from triangle listing complexities.

Findings

Algorithm lists all 4-cycles in ( ilde{O}( ext{min}(n^2,m^{4/3})+t)) time.

The bound is conditionally tight based on recent lower bounds related to the 3-SUM conjecture.

The work distinguishes 4-cycle listing complexity from triangle listing, showing different computational bounds.

Abstract

In this note we present an algorithm that lists all $4$-cycles in a graph in time $\tilde{O}(\min(n^2,m^{4/3})+t)$ where $t$ is their number. Notably, this separates $4$-cycle listing from triangle-listing, since the latter has a $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ lower bound under the $3$-SUM Conjecture. Our upper bound is conditionally tight because (1) $O(n^2,m^{4/3})$ is the best known bound for detecting if the graph has any $4$-cycle, and (2) it matches a recent $(\min(n^3,m^{3/2})+t)^{1-o(1)}$ $3$-SUM lower bound for enumeration algorithms. The latter lower bound was proved very recently by Abboud, Bringmann, and Fischer [arXiv, 2022] and independently by Jin and Xu [arXiv, 2022]. In an independent work, Jin and Xu [arXiv, 2022] also present an algorithm with the same time bound.