On the second eigenvalue of random regular graphs
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Polynomial time approximation schemes for dense instances of NP -hard problems
Journal of Computer and System Sciences
On Some Tighter Inapproximability Results (Extended Abstract)
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
The regularity lemma and approximation schemes for dense problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Clustering with Qualitative Information
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On the Edge-Expansion of Graphs
Combinatorics, Probability and Computing
Correlation Clustering: maximizing agreements via semidefinite programming
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Machine Learning
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximating the Cut-Norm via Grothendieck's Inequality
SIAM Journal on Computing
Complexity of approximating bounded variants of optimization problems
Theoretical Computer Science - Foundations of computation theory (FCT 2003)
Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph
Graphs and Combinatorics
IEEE Transactions on Knowledge and Data Engineering
Significance-Driven Graph Clustering
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
A Scalable Multilevel Algorithm for Graph Clustering and Community Structure Detection
Algorithms and Models for the Web-Graph
Multi-level Algorithms for Modularity Clustering
SEA '09 Proceedings of the 8th International Symposium on Experimental Algorithms
Asymptotic modularity of some graph classes
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Mixing local and global information for community detection in large networks
Journal of Computer and System Sciences
On a connection between small set expansions and modularity clustering
Information Processing Letters
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Given a graph of interactions, a module (also called a community or cluster) is a subset of nodes whose fitness is a function of the statistical significance of the pairwise interactions of nodes in the module. The topic of this paper is a model-based community finding approach, commonly referred to as modularity clustering, that was originally proposed by Newman (Leicht and Newman, 2008 [25]) and has subsequently been extremely popular in practice (e.g., see Agarwal and Kempe, 2008 [1], Guimer'a et al., 2007 [20], Newman, 2006 [28], Newman and Girvan, 2004 [30], Ravasz et al., 2002 [32]). Various heuristic methods are currently employed for finding the optimal solution. However, as observed in Agarwal and Kempe (2008) [1], the exact computational complexity of this approach is still largely unknown. To this end, we initiate a systematic study of the computational complexity of modularity clustering. Due to the specific quadratic nature of the modularity function, it is necessary to study its value on sparse graphs and dense graphs separately. Our main results include a (1+@e)-inapproximability for dense graphs and a logarithmic approximation for sparse graphs. We make use of several combinatorial properties of modularity to get these results. These are the first non-trivial approximability results beyond the NP-hardness results in Brandes et al. (2007) [10].