CIAC 2003: Algorithms and Complexity pp 201-212

# The Complexity of Detecting Fixed-Density Clusters

• Klaus Holzapfel
• Sven Kosub
• Moritz G. Maaß
• Hanjo Täubig
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2653)

## 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 $$\gamma = 2 + 0\left( {\frac{1} {k}} \right)$$ .

### Keywords

Density-based clustering computational complexity graph algorithms fixed-parameter problems

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## Authors and Affiliations

• Klaus Holzapfel
• 1
• Sven Kosub
• 1
• Moritz G. Maaß
• 1
• Hanjo Täubig
• 1
1. 1.Fakultät für InformatikTechnische Universität MünchenGarching b. MünchenGermany