Approximability of modularity clustering and related results

Abstract

OF THE DISSERTATION Approximability of Modularity Clustering and related results by DEVENDRA J. DESAI Dissertation Director: Mario Szegedy For the problem of Modularity Clustering, first introduced by Newman and Girvan in 2004, we are given a graph and the goal is to partition the vertex set into an unknown number of clusters such that we maximize a certain objective function which evaluates the fitness of the clusters. This fitness function measures the statistically surprising distribution of edges between different clusters and in the clusters themselves. Despite having found widespread popularity in the fields of biology and social sciences, this problem is known to be NP-hard and up till the work in this thesis, only heuristics were known. In this thesis, we initiate a study of the approximability of modularity clustering. We give the first approximation algorithms and the first hardness of approximation results for the problem. In doing so, we employ various techniques like semidefinite programming and the regularity lemma. Our main results include a factor 1.0009 inapproximability for dense graphs and a logarithmic (in the