On the Complexity of Nondeterministically Testable Hypergraph Parameters

The paper proves the equivalence of the notions of nondeterministic and deterministic parameter testing for uniform dense hypergraphs of arbitrary order. It generalizes the result previously known only for the case of simple graphs. By a similar method we establish also the equivalence between nondeterministic and deterministic hypergraph property testing, answering the open problem in the area. We introduce a new notion of a cut norm for hypergraphs of higher order, and employ regularity techniques combined with the ultralimit method.Comment: 33 page

On Approximability, Convergence, and Limits of CSP Problems

This thesis studies dense constraint satisfaction problems (CSPs), and other related optimization and decision problems that can be phrased as questions regarding parameters or properties of combinatorial objects such as uniform hypergraphs. We concentrate on the information that can be derived from a very small substructure that is selected uniformly at random. In this thesis, we present a unified framework on the limits of CSPs in the sense of the convergence notion of Lovasz-Szegedy that depends only on the remarkable connection between graph sequences and exchangeable arrays established by Diaconis-Janson. In particular, we formulate and prove a representation theorem for compact colored r-uniform directed hypergraphs and apply this to rCSPs. We investigate the sample complexity of testable r-graph parameters, and discuss a generalized version of ground state energies (GSE) and demonstrate that they are efficiently testable. The GSE is a term borrowed from statistical physics that stands for a generalized version of maximal multiway cut problems from complexity theory, and was studied in the dense graph setting by Borgs et al. A notion related to testing CSPs that are defined on graphs, the nondeterministic property testing, was introduced by Lovasz-Vesztergombi, which extends the graph property testing framework of Goldreich-Goldwasser-Ron in the dense graph model. In this thesis, we study the sample complexity of nondeterministically testable graph parameters and properties and improve existing bounds by several orders of magnitude. Further, we prove the equivalence of the notions of nondeterministic and deterministic parameter and property testing for uniform dense hypergraphs of arbitrary rank, and provide the first effective upper bound on the sample complexity in this general case

A characterization of testable hypergraph properties

We provide a combinatorial characterization of all testable properties of $k$-graphs (i.e. $k$-uniform hypergraphs). Here, a $k$-graph property $\mathbf{P}$ is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability $2/3$ between $k$-graphs that satisfy $\mathbf{P}$ and those that are far from satisfying $\mathbf{P}$. For the $2$-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the $k$-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the $3$-graph setting.Comment: 82 pages; extended abstract of this paper appears in FOCS 201

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field