### Finding and Counting Cliques with cliquer

For a graph $G$, a clique is a set $S \subseteq V(G)$ such that all pairs of vertices in $S$ are adjacent in $G$. Determining the maximum size of a clique in a graph is a classic NP-Complete problem, but sometimes you just need to find or count cliques anyway.

Today, we discuss the cliquer algorithm, as designed by Niskanen and Östergård. Their very efficient implementation is available on their cliquer homepage. The entire implementation is very succinct and can be used as a library. In fact, Sage uses cliquer for its backend for clique algorithms. The main algorithm computes the maximum (weighted) clique and also can count maximum or maximal (weighted) cliques. We focus on the unweighted case.
Read the rest of this entry »