Finance

Operating system

Algorithm

Combinatorics

Dense graph

Graph

Economics

Parallel computing

Line graph

Clique

Listing (finance)

Mathematics

Computer science

Pathwidth

Theoretical computer science

Subroutine

Motivated by recent studies in the data mining community which require to efficiently list all k-cliques, we revisit the iconic algorithm of Chiba and Nishizeki and develop the most efficient parallel algorithm for such a problem. Our theoretical analysis provides the best asymptotic upper bound on the running time of our algorithm for the case when the input graph is sparse. Our experimental evaluation on large real-world graphs shows that our parallel algorithm is faster than state-of-the-art algorithms, while boasting an excellent degree of parallelism. In particular, we are able to list all k-cliques (for any k) in graphs containing up to tens of millions of edges as well as all $10$-cliques in graphs containing billions of edges, within a few minutes and a few hours respectively. Finally, we show how our algorithm can be employed as an effective subroutine for finding the k-clique core decomposition and an approximate k-clique densest subgraphs in very large real-world graphs.

Listing k-cliques in Sparse Real-World Graphs*