The New Deal for Communities experience: a final assessment - The New Deal for Communities evaluation: Final report – Volume 7

We show that for all integers $t\geq 8$ and arbitrarily small $\epsilon>0$, there exists a graph property $\Pi$ (which depends on $\epsilon$) such that $\epsilon$-testing $\Pi$ has non-adaptive query complexity $Q=\~{\Theta}(q^{2-2/t})$, where $q=\~{\Theta}(\epsilon^{-1})$ is the adaptive query complexity. This resolves the question of how beneficial adaptivity is, in the context of proximity-dependent properties (\cite{benefits-of-adaptivity}). This also gives evidence that the canonical transformation of Goldreich and Trevisan (\cite{canonical-testers}) is essentially optimal when converting an adaptive property tester to a non-adaptive property tester. To do so, we provide optimal adaptive and non-adaptive testers for the combined property of having maximum degree $O(\epsilon N)$ and being a \emph{blow-up collection} of an arbitrary base graph $H$.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Dense-Graph Model, Adaptive vs Nonadaptive Queries, Hierarchy Theore

Constructing dense graphs with sublinear Hadwiger number

Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by Thomason over several articles. We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property. This leads to the study of a fractional variant of the clique minor number, which may be of independent interest.Comment: 10 page

A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing

Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that for each $\epsilon>0$ the vertex set can be equitably partitioned into a bounded number of parts (in terms of $\epsilon$ and the complexity) so that all but an $\epsilon$-fraction of the $k$-tuples of parts are homogeneous. We prove that the number of parts can be taken to be polynomial in $1/\epsilon$. Our improved regularity lemma can be applied to geometric problems and to the following general question on property testing: is it possible to decide, with query complexity polynomial in the reciprocal of the approximation parameter, whether a hypergraph has a given hereditary property? We give an affirmative answer for testing typical hereditary properties for semi-algebraic hypergraphs of bounded complexity

Many TT copies in HH-free graphs

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) $ex(n,K_3,C_5) \leq (1+o(1)) \frac{\sqrt 3}{2} n^{3/2},$ (ii) For any fixed $m$, $s \geq 2m-2$ and $t \geq (s-1)!+1$, $ex(n,K_m,K_{s,t})=\Theta(n^{m-\binom{m}{2}/s})$ and (iii) For any two trees $H$ and $T$, $ex(n,T,H) =\Theta (n^m)$ where $m=m(T,H)$ is an integer depending on $H$ and $T$ (its precise definition is given in Section 1). The first result improves (slightly) an estimate of Bollob\'as and Gy\H{o}ri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques

Aspects of energy requirements for rock drilling

Development of laboratory rock breakage techniques to relate energy and surface area produced by slow compression, drop hammer and stamp mill. A detailed study of laboratory rotary-percussive drilling in a wide range of rocks under different conditions, with the collection of drill cuttings and measurement of the drill parameters. The correlation of drill parameters with rock indices by energy concepts and the developed empirical formula. Field rotary-percussive drilling studies and collection of drill cuttings on the basis of laboratory analysis

A Dirac type result on Hamilton cycles in oriented graphs

We show that for each \alpha>0 every sufficiently large oriented graph G with \delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the stronger result that G is still Hamiltonian if \delta(G)+\delta^+(G)+\delta^-(G)\geq 3|G|/2 + \alpha |G|. Up to the term \alpha |G| this confirms a conjecture of H\"aggkvist. We also prove an Ore-type theorem for oriented graphs.Comment: Added an Ore-type resul

An Algorithm to Compute the Topological Euler Characteristic, Chern-Schwartz-MacPherson Class and Segre Class of Projective Varieties

Let $V$ be a closed subscheme of a projective space $\mathbb{P}^n$. We give an algorithm to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of $V$. The algorithm can be implemented using either symbolic or numerical methods. The algorithm is based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Using this result and known formulas for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps we give algorithms to compute the Chern-Schwartz-MacPherson class and Segre class of a projective variety. Since the Euler characteristic of $V$ is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of $V$ our algorithm also computes the Euler characteristic $\chi(V)$. Relationships between the algorithm developed here and other existing algorithms are discussed. The algorithm is tested on several examples and performs favourably compared to current algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics

Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gr\"obner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo-Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind-Mertens formula for the content of the product of two polynomials.Comment: 36 pages, 9 figures. In the updated version, section 7: "Final remarks and open problems" is new; the introduction was updated accordingly. References update