16 research outputs found

    Overlap properties of geometric expanders

    Get PDF
    The {\em overlap number} of a finite (d+1)(d+1)-uniform hypergraph HH is defined as the largest constant c(H)(0,1]c(H)\in (0,1] such that no matter how we map the vertices of HH into Rd\R^d, there is a point covered by at least a c(H)c(H)-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence {Hn}n=1\{H_n\}_{n=1}^\infty of arbitrarily large (d+1)(d+1)-uniform hypergraphs with bounded degree, for which infn1c(Hn)>0\inf_{n\ge 1} c(H_n)>0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d+1)(d+1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c=c(d)c=c(d). We also show that, for every dd, the best value of the constant c=c(d)c=c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d+1)(d+1)-uniform hypergraphs with nn vertices, as nn\rightarrow\infty. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any dd and any ϵ>0\epsilon>0, there exists K=K(ϵ,d)d+1K=K(\epsilon,d)\ge d+1 satisfying the following condition. For any kKk\ge K, for any point qRdq \in \mathbb{R}^d and for any finite Borel measure μ\mu on Rd\mathbb{R}^d with respect to which every hyperplane has measure 00, there is a partition Rd=A1Ak\mathbb{R}^d=A_1 \cup \ldots \cup A_{k} into kk measurable parts of equal measure such that all but at most an ϵ\epsilon-fraction of the (d+1)(d+1)-tuples Ai1,,Aid+1A_{i_1},\ldots,A_{i_{d+1}} have the property that either all simplices with one vertex in each AijA_{i_j} contain qq or none of these simplices contain qq

    Nonlinear spectral calculus and super-expanders

    Get PDF
    Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

    Maximum gradient embeddings and monotone clustering

    Full text link
    Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica
    corecore