On Clustering Detection Based on a Quadratic Program in Hypergraphs

A proper cluster is usually defined as maximally coherent groups from a set of objects using pairwise or more complicated similarities. In general hypergraphs, clustering problem refers to extraction of subhypergraphs with a higher internal density, for instance, maximal cliques in hypergraphs. The determination of clustering structure within hypergraphs is a significant problem in the area of data mining. Various works of detecting clusters on graphs and uniform hypergraphs have been published in the past decades. Recently, it has been shown that the maximum 1,2 -clique size in 1,2 -hypergraphs is related to the global maxima of a certain quadratic program based on the structure of the given nonuniform hypergraphs. In this paper, we first extend this result to relate strict local maxima of this program to certain maximal cliques including 2-cliques or 1,2 -cliques. We also explore the connection between edge-weighted clusters and strictly local optimum solutions of a class of polynomials resulting from nonuniform 1,2 -hypergraphs.

Local Maxima of Quadratic Boolean Functions

How many strict local maxima can a real quadratic function on {0, 1}n have? Holzman conjectured a maximum of $\binom{n }{ \lfloor n/2 \rfloor}$. The aim of this paper is to prove this conjecture. Our approach is via a generalization of Sperner's theorem that may be of independent interest.