Dense clusters in hypergraphs

In this paper we solve the problem of finding in a given weighted hypergraph a subhypergraph with a maximum possible density. We introduce the notion of a support matrix and prove that the density of an optimal subhypergraph is equal to $\mathrm{\mid }{A}^{T}A\mathrm{\mid }$ for an optimal support matrix $A$. Alternatively, the maximum density of a subhypergraph is equal to the solution of a minimax problem for column sums of support matrices. We introduce the spectral decomposition of a hypergraph and show that it is a significant refinement of the Dulmage-Mendelsohn decomposition. Our theoretical results yield an efficient algorithm for finding the maximum density subhypergraph and more generally, the spectral decomposition for a given weighted hypergraph.