Diszkrét matematika = Discrete mathematics

A pályázat résztvevői igen aktívak voltak a 2006-2008 években. Nemcsak sok eredményt értek el, miket több mint 150 cikkben publikáltak, eredményesen népszerűsítették azokat. Több mint 100 konferencián vettek részt és adtak elő, felerészben meghívott, vagy plenáris előadóként. Hagyományos gráfelmélet Több extremális gráfproblémát oldottunk meg. Új eredményeket kaptunk Ramsey számokról, globális és lokális kromatikus számokról, Hamiltonkörök létezéséséről. a crossig numberről, gráf kapacitásokról és kizárt részgráfokról. Véletlen gráfok, nagy gráfok, regularitási lemma Nagy gráfok "hasonlóságait" vizsgáltuk. Különféle metrikák ekvivalensek. Űj eredeményeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. Hipergráfok, egyéb kombinatorika Új Sperner tipusú tételekte kaptunk, aszimptotikusan meghatározva a halmazok max számát bizonyos kizárt struktőrák esetén. Több esetre megoldottuk a kizárt hipergráf problémát is. Elméleti számítástudomány Új ujjlenyomat kódokat és bioinformatikai eredményeket kaptunk. | The participants of the project were scientifically very active during the years 2006-2008. They did not only obtain many results, which are contained in their more than 150 papers appeared in strong journals, but effectively disseminated them in the scientific community. They participated and gave lectures in more than 100 conferences (with multiplicity), half of them were plenary or invited talks. Traditional graph theory Several extremal problems for graphs were solved. We obtained new results for certain Ramsey numbers, (local and global) chromatic numbers, existence of Hamiltonian cycles crossing numbers, graph capacities, and excluded subgraphs. Random graphs, large graphs, regularity lemma The "similarities" of large graphs were studied. We show that several different definitions of the metrics (and convergence) are equivalent. Several new results like the Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit were proved Hypergraphs, other combinatorics New Sperner type theorems were obtained, asymptotically determining the maximum number of sets in a family of subsets with certain excluded configurations. Several cases of the excluded hypergraph problem were solved. Theoretical computer science New fingerprint codes and results in bioinformatics were found

Extremal Mechanisms for Local Differential Privacy

Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and $f$-divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime.Comment: 52 pages, 10 figures in JMLR 201

On the Complexity of Existential Positive Queries

We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted to fall into the set; a natural question is then to classify which sentence sets are tractable and which are intractable. With respect to fixed-parameter tractability, we give a general theorem that reduces this classification question to the corresponding question for primitive positive logic, for a variety of representations of structures. This general theorem allows us to deduce that an existential positive sentence set having bounded arity is fixed-parameter tractable if and only if each sentence is equivalent to one in bounded-variable logic. We then use the lens of classical complexity to study these fixed-parameter tractable sentence sets. We show that such a set can be NP-complete, and consider the length needed by a translation from sentences in such a set to bounded-variable logic; we prove superpolynomial lower bounds on this length using the theory of compilability, obtaining an interesting type of formula size lower bound. Overall, the tools, concepts, and results of this article set the stage for the future consideration of the complexity of model checking on more expressive logics

Functional declarative language design and predicate calculus: A practical approach

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Truthful Linear Regression

We consider the problem of fitting a linear model to data held by individuals who are concerned about their privacy. Incentivizing most players to truthfully report their data to the analyst constrains our design to mechanisms that provide a privacy guarantee to the participants; we use differential privacy to model individuals' privacy losses. This immediately poses a problem, as differentially private computation of a linear model necessarily produces a biased estimation, and existing approaches to design mechanisms to elicit data from privacy-sensitive individuals do not generalize well to biased estimators. We overcome this challenge through an appropriate design of the computation and payment scheme.Comment: To appear in Proceedings of the 28th Annual Conference on Learning Theory (COLT 2015