Algorithm for finding cliques

Question

Given an arbitrary undirected graph $G = (V,E)$, I am interested in a low-polynomial time algorithm which can find several moderately large (ideally $O(n^\epsilon)$ vertices per clique for $\epsilon > 0$) cliques in $G$, provided they exist.

I am aware that the problem of finding the maximal clique is NP-hard. For my purposes, however, I don't need maximality per say. Rather, it would be helpful to have an algorithm which runs relatively quickly and with high probability finds some nontrivial cliques.

I am aware that for fixed $k$, finding a $k$-clique if it exists can be done in $n^k$ time by simply testing all subsets of $k$ vertices. This algorithm appears to be of limited use because of the exponential growth in $k$.

Does there exist an algorithm or multiple which can, with high probability, find multiple moderately large cliques in a general undirected graph?

Answer 1

Suppose that there exists a polytime algorithm which finds a clique of size $n^\epsilon$ if there is any. Then you can approximate maximum clique to within $n^{1-\epsilon}$ (assuming $\epsilon \leq 1/2$), which is NP-hard by a classical result of Håstad. Indeed, run your algorithm (stopping it if it exceeds its advertized running time), and check whether its output is a clique of size at least $n^\epsilon$. If so, output $n^\epsilon$, otherwise output $1$.

You might want to look into the planted clique problem, in which the goal is to find a clique "planted" inside a uniformly random graph. This is possible in polytime (with high probability) as long as the clique is large enough, $\Omega(\sqrt{n})$. Your case may be different, but similar ideas might work in practice.