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WWW '99 Proceedings of the eighth international conference on World Wide Web
Greedily finding a dense subgraph
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APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
When clusters meet partitions: new density-based methods for circuit decomposition
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On the densest k-subgraph problems
On the densest k-subgraph problems
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Operations Research Letters
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Discrete Applied Mathematics
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Proceedings of the 16th international conference on World Wide Web
On the NP-Completeness of some graph cluster measures
SOFSEM'06 Proceedings of the 32nd conference on Current Trends in Theory and Practice of Computer Science
Computer Science Review
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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 γ = 2+Ω(1/k1-ɛ) for some ɛ 0 and has a polynomial-time algorithm for γ = 2+O(1/k).