The Complexity of Detecting Fixed-Density Clusters

Abstract

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 γ : ℕ → ℚ₊ be any density function, i.e., γ is computable in polynomial time and satisfies γ(k) ≤ k − 1 for all k ∈ ℕ. Then γ-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 which has average degree at least γ(k). For γ(k) = k − 1, 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 γ(k) = 2. We ask for the possible functions γ such that γ-Cluster remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: γ-Cluster is NP-complete if $\gamma = 2 + \Omega \left( {\frac{1} {{k^1 - \mathcal{E}}}} \right)$ for some ε > 0 and has a polynomial-time algorithm for $\gamma = 2 + 0\left( {\frac{1} {k}} \right)$ .

Research of the third and of the fourth author supported by DFG, grant Ma 870/5-1 (Leibnizpreis Ernst W. Mayr).

Research supported by DFG (Deutsche Forschungsgemeinschaft), grant Ma 870/6-1 (SPP 1126 Algorithmik großer und komplexer Netzwerke).