Tailored vertex ordering for faster triangle listing in large graphs

TL;DR

This paper introduces new cost functions and heuristics for vertex ordering in large graphs, significantly improving the speed of triangle listing algorithms by up to 38%.

Contribution

It studies the precise cost of triangle listing algorithms, introduces NP-hardness proofs, and proposes heuristics for better vertex orderings to enhance performance.

Findings

Heuristics reduce triangle listing time by 38% on large datasets.

Cost minimization is NP-hard, indicating the complexity of optimal ordering.

Including ordering time, heuristics still achieve a 16% speedup.

Abstract

Listing triangles is a fundamental graph problem with many applications, and large graphs require fast algorithms. Vertex ordering allows the orientation of edges from lower to higher vertex indices, and state-of-the-art triangle listing algorithms use this to accelerate their execution and to bound their time complexity. Yet, only basic orderings have been tested. In this paper, we show that studying the precise cost of algorithms instead of their bounded complexity leads to faster solutions. We introduce cost functions that link ordering properties with the running time of a given algorithm. We prove that their minimization is NP-hard and propose heuristics to obtain new orderings with different trade-offs between cost reduction and ordering time. Using datasets with up to two billion edges, we show that our heuristics accelerate the listing of triangles by an average of 38% when the ordering is already given as an input, and 16% when the ordering time is included.