The complexity of detecting fixed-density clusters Original Research Article

We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let image be any density function, i.e., image is computable in polynomial time and satisfies image for all image. Then image-CLUSTER is the problem of deciding, given an undirected graph G and a natural number k, whether there is a subgraph of G on k vertices that has average degree at least image. For image, this problem is the same as the well-known CLIQUE problem, and thus NP-complete. In contrast to this, the problem is known to be solvable in polynomial time for image. We ask for the possible functions image such that image-CLUSTER remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: image CLUSTER is NP-complete if image for some image and has a polynomial-time algorithm for image. The algorithm also shows that for image, image-CLUSTER is solvable in subexponential time image.