Parallel Algorithms for Finding Large Cliques in Sparse Graphs

TL;DR

This paper introduces a parallel algorithm for listing large cliques in sparse graphs, improving efficiency by novel pruning and approximation techniques, especially for larger cliques and graphs with low community degeneracy.

Contribution

It presents a new parallel clique listing algorithm with improved work bounds and a low-depth approximation method based on community degeneracy.

Findings

Enhanced asymptotic performance for large cliques in sparse graphs.

Exponential work improvement for cliques smaller than the graph's degeneracy.

Low-depth algorithm leveraging community degeneracy approximation.

Abstract

We present a parallel k-clique listing algorithm with improved work bounds (for the same depth) in sparse graphs with low degeneracy or arboricity. We achieve this by introducing and analyzing a new pruning criterion for a backtracking search. Our algorithm has better asymptotic performance, especially for larger cliques (when k is not constant), where we avoid the straightforwardly exponential runtime growth with respect to the clique size. In particular, for cliques that are a constant factor smaller than the graph's degeneracy, the work improvement is an exponential factor in the clique size compared to previous results. Moreover, we present a low-depth approximation to the community degeneracy (which can be arbitrarily smaller than the degeneracy). This approximation enables a low depth clique listing algorithm whose runtime is parameterized by the community degeneracy.