Fourier analysis, linear programming, and densities of distance avoiding sets in {RnR^n}

In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances

New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l_3^p$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. These bounds give strong indications that some of the lattice packings of superballs found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$. We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization

A recursive theta body for hypergraphs

The theta body of a graph, introduced by Grötschel, Lovász, and Schrijver (in 1986), is a tractable relaxation of the independent-set polytope derived from the Lovász theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel’s theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube

New reduction techniques for the group Steiner tree problem

The group Steiner tree problem consists of, given a graph $G$, a collection $\mathcal{R}$ of subsets of $V(G)$, and a positive cost $c_e$ for each edge $e$ of $G$, finding a minimum-cost tree in $G$ that contains at least one vertex from each $R \in \mathcal{R}$. We call the sets in $\mathcal{R}$ groups. The well-known Steiner tree problem is the special case of the group Steiner tree problem in which each set in $\mathcal{R}$ is unitary. In this paper, we present a general reduction test designed to remove group vertices, that is, vertices belonging to some group. Through the use of these tests we can conclude that a given group vertex can be considered a nonterminal and hence can be removed from its group. We also present some computational results on instances from SteinLib [T. Koch, A. Martin, and S. Voss, SteinLib: An updated library on Steiner tree problems in graphs, in Steiner Trees in Industry, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 285–325]

Some formulations for the group Steiner tree problem

AbstractThe group Steiner tree problem consists of, given a graph G, a collection R of subsets of V(G) and a cost c(e) for each edge of G, finding a minimum-cost subtree that connects at least one vertex from each R∈R. It is a generalization of the well-known Steiner tree problem that arises naturally in the design of VLSI chips. In this paper, we study a polyhedron associated with this problem and some extended formulations. We give facet defining inequalities and explore the relationship between the group Steiner tree problem and other combinatorial optimization problems

A quantitative version of Steinhaus’ theorem for compact, connected, rank-one symmetric spaces

Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$, $d_2$, ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances

Fourier analysis, linear programming, and densities of distance avoiding sets in {RnR^n}

In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances

The positive semidefinite Grothendieck problem with rank constraint

Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint is (SDP_n) maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, >..., x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 - \Theta(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial time algorithm which approximates SDP_n with a ratio greater than \gamma(n). We improve the approximation ratio of the best known polynomial time algorithm for SDP_1 from 2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we determine the optimal constant of the positive semidefinite case of a generalized Grothendieck inequality