Krzysztof Choromanski

Forcing large transitive subtournaments

The Erd?s-Hajnal Conjecture states that for every given H there exists a constant c ( H ) 0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least | V ( G ) | c ( H ) . The conjecture is still open. However some time ago its directed version was proved to be equivalent to the original one. In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. Both the directed and the undirected versions of the conjecture are known to be true for small graphs (or tournaments), and there are operations (the so-called substitution operations) allowing to build bigger graphs (or tournaments) for which the conjecture holds. In this paper we prove the conjecture for an infinite class of tournaments that is not obtained by such operations. We also show that the conjecture is satisfied by every tournament on at most 5 vertices.

EH-suprema of tournaments with no nontrivial homogeneous sets

A celebrated unresolved conjecture of Erdos and Hajnal states that for every undirected graph H there exists ? ( H ) 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least n ? ( H ) .The conjecture has directed equivalent version stating that for every tournament H there exists ? ( H ) 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n ? ( H ) . For a fixed tournament H, define ? ( H ) to be the supremum of all ? for which the following holds: for some n 0 and every n n 0 every tournament with n ? n 0 vertices not containing H as a subtournament has a transitive subtournament of size at least n ? . We call ? ( H ) the EH-supremum of H. The Erdos-Hajnal conjecture is true if and only if ? ( H ) 0 for every H. If the conjecture is false then the smallest counterexample has no nontrivial so-called homogeneous sets (to be defined below). Therefore of interest are EH-suprema of those tournaments. In 5] it was proven that there exists a constant ? 0 such that ? ( H ) ? 4 h ( 1 + ? log ? ( h ) h ) for almost every h-vertex tournament H. However this result does not say anything about ? ( H ) for an arbitrarily chosen tournament with no nontrivial homogeneous sets. We address that problem in this paper, proving that there exists C 0 such that every h-vertex tournament H with no nontrivial homogeneous sets satisfies ? ( H ) ? C log ? ( h ) h . We will also give upper bounds on sizes of families of h-vertex tournaments with big EH-suprema. In 1] Alon, Pach and Solymosi proposed a procedure that produces bigger graphs satisfying the conjecture from smaller ones. All graphs obtained in such a way have nontrivial homogeneous sets. For a long time that was the only method to obtain infinite families of graphs satisfying the conjecture. Recently Berger, the author and Chudnovsky (see 2]) constructed a new infinite family of tournaments (so-called galaxies, to be defined below) that satisfies the conjecture and with no nontrivial homogeneous sets. Therefore it cannot be obtained by the procedure described in 1]. In this paper we construct a new infinite family of tournaments satisfying the conjecture - the family of so-called constellations (to be defined below). These results extend the results of 2] since every galaxy is a constellation.

The power of the dinur-nissim algorithm: breaking privacy of statistical and graph databases

A few years ago, Dinur and Nissim (PODS, 2003) proposed an algorithm for breaking database privacy when statistical queries are answered with a perturbation error of magnitude o(√n) for a database of size n. This negative result is very strong in the sense that it completely reconstructs Ω(n) data bits with an algorithm that is simple, uses random queries, and does not put any restriction on the perturbation other than its magnitude. Their algorithm works for a model where the database consists of bits, and the statistical queries asked by the adversary are sum queries for a subset of locations. In this paper we extend the attack to work for much more general settings in terms of the type of statistical query allowed, the database domain, and the general tradeoff between perturbation and privacy. Specifically, we prove: For queries of the type ∑in=1 φixi; where φ_{i} are i.i.d. and with a finite third moment and positive variance (this includes as a special case the sum queries of Dinur-Nissim and several subsequent extensions), we prove that the quadratic relation between the perturbation and what the adversary can reconstruct holds even for smaller perturbations, and even for a larger data domain. If φi is Gaussian, Poissonian, or bounded and of positive variance, this holds for arbitrary data domains and perturbation; for other φi this holds as long as the domain is not too large and the perturbation is not too small. A positive result showing that for a sum query the negative result mentioned above is tight. Specifically, we build a distribution on bit databases and an answering algorithm such that any adversary who wants to recover a little more than the negative result above allows, will not succeed except with negligible probability. We consider a richer class of summation queries, focusing on databases representing graphs, where each entry is an edge, and the query is a structural function of a subgraph. We show an attack that recovers a big portion of the graph edges, as long as the graph and the function satisfy certain properties. The attacking algorithms in both our negative results are straight-forward extensions of the Dinur-Nissim attack, based on asking φ-weighted queries or queries choosing a subgraph uniformly at random. The novelty of our work is in the analysis, showing that this simple attack is much more powerful than was previously known, as well as pointing to possible limits of this approach and putting forth new application domains such as graph problems (which may occur in social networks, Internet graphs, etc). These results may find applications not only for breaking privacy, but also in the positive direction, for recovering complicated structure information using inaccurate estimates about its substructures.

Differentially-Private Learning of Low Dimensional Manifolds

In this paper, we study the problem of differentially-private learning of low dimensional manifolds embedded in high dimensional spaces. The problems one faces in learning in high dimensional spaces are compounded in differentially-private learning. We achieve the dual goals of learning the manifold while maintaining the privacy of the dataset by constructing a differentially-private data structure that adapts to the doubling dimension of the dataset. Our differentially-private manifold learning algorithm extends random projection trees of Dasgupta and Freund. A naive construction of differentially-private random projection trees could involve queries with high global sensitivity that would affect the usefulness of the trees. Instead, we present an alternate way of constructing differentially-private random projection trees that uses low sensitivity queries that are precise enough for learning the low dimensional manifolds. We prove that the size of the tree depends only on the doubling dimension of the dataset and not its extrinsic dimension.

Excluding pairs of tournaments

The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph

Differentially-private learning of low dimensional manifolds

H

An Optimal Online Algorithm For Retrieving Heavily Perturbed Statistical Databases In The Low-Dimensional Querying Model

there exists a constant

Scale-Free Graph with Preferential Attachment and Evolving Internal Vertex Structure

c(H)>0

Explaining How a Deep Neural Network Trained with End-to-End Learning Steers a Car.

such that every graph

Orthogonal Random Features

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