Breaking the hegemony of the triangle method in clique detection

TL;DR

This paper introduces a new approach using multi-dimensional matrix products for detecting and counting cliques in graphs, challenging the dominance of the triangle-based method especially for K4 and K5 cliques.

Contribution

It presents an alternative clique detection method based on multi-dimensional matrix products, offering comparable or better efficiency than traditional triangle-based algorithms.

Findings

Multi-dimensional matrix product method is as efficient as triangle detection for K4 and K5.

Proposed methods provide new tools for clique detection and counting.

Complexity analysis of multi-dimensional matrix products is included.

Abstract

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of $K_4$ and $K_5.$ The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting $K_r$ copies, again time efficient for $r\in \{ 4,\ 5 \}$.