Local And Global Colorability of Graphs

It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tilde{\Theta}\left(n ^ {\frac{1}{r+1}} \right)$ (that is, $n^\frac{1}{r+1}$ up to a factor poly-logarithmic in $n$). The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random $n$-vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius $r$ in it are $2$-degenerate

Biased orientation games

We study biased {\em orientation games}, in which the board is the complete graph $K_n$, and Maker and Breaker take turns in directing previously undirected edges of $K_n$. At the end of the game, the obtained graph is a tournament. Maker wins if the tournament has some property $\mathcal P$ and Breaker wins otherwise. We provide bounds on the bias that is required for a Maker's win and for a Breaker's win in three different games. In the first game Maker wins if the obtained tournament has a cycle. The second game is Hamiltonicity, where Maker wins if the obtained tournament contains a Hamilton cycle. Finally, we consider the $H$-creation game, where Maker wins if the obtained tournament has a copy of some fixed graph $H$

Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large $d$-dimensional array $A$ and a fixed $d$-dimensional pattern $P$ over a finite alphabet, we say that $A$ is $P$-free if it does not contain a copy of the forbidden pattern $P$ as a consecutive subarray. The distance of $A$ to $P$-freeness is the fraction of entries of $A$ that need to be modified to make it $P$-free. For any $\epsilon \in [0,1]$ and any large enough pattern $P$ over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least $\epsilon$ and the case that it is at most $a_d \epsilon$, with query complexity and running time $c_d \epsilon^{-1}$, where $a_d < 1$ and $c_d$ depend only on $d$. To analyze the testers we establish several combinatorial results, including the following $d$-dimensional modification lemma, which might be of independent interest: for any large enough pattern $P$ over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array $A$ containing a copy of $P$, one can delete this copy by modifying one of its locations without creating new $P$-copies in $A$. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns

Las etnicidades judías en Israel

La fundación del Estado de Israel se acompañó de un proyecto social, cultural y político destinado a asegurar la creación de una nación homogénea. La clase media proveniente de Europa oriental y su visión laica de la cultura judía en Israel ha tenido unaThe founding of the state of Israel was accompanied by a social, cultural and political project, which aim was to secure the establishment of a homogenous nation. The middle class, which originated from Eastern Europe, and its secular vision of the stat

Testing Local Properties of Arrays

We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of d-dimensional arrays f:[n]^d -> Sigma is k-local if it can be defined by a family of k x ... x k forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and submodularity are 2-local; convexity is (usually) 3-local; and many typical problems in computational biology and computer vision involve o(n)-local properties. In this work, we present a generic approach to test all local properties of arrays over any finite (and not necessarily bounded size) alphabet. We show that any k-local property of d-dimensional arrays is testable by a simple canonical one-sided error non-adaptive epsilon-test, whose query complexity is O(epsilon^{-1}k log{(epsilon n)/k}) for d = 1 and O(c_d epsilon^{-1/d} k * n^{d-1}) for d > 1. The queries made by the canonical test constitute sphere-like structures of varying sizes, and are completely independent of the property and the alphabet Sigma. The query complexity is optimal for a wide range of parameters: For d=1, this matches the query complexity of many previously investigated local properties, while for d > 1 we design and analyze new constructions of k-local properties whose one-sided non-adaptive query complexity matches our upper bounds. For some previously studied properties, our method provides the first known sublinear upper bound on the query complexity

Limits of Ordered Graphs and their Applications

The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in analytic language. We extend the theory of graph limits to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which we call an \emph{orderon}. As a special case, this yields limit objects for matrices whose rows and columns are ordered, and for dynamic graphs that expand (via vertex insertions) over time. Along the way, we devise an ordered locality-preserving variant of the cut distance between ordered graphs, showing that two graphs are close with respect to this distance if and only if they are similar in terms of their ordered subgraph frequencies. We show that the space of orderons is compact with respect to this distance notion, which is key to a successful analysis of combinatorial objects through their limits. We derive several applications of the ordered limit theory in extremal combinatorics, sampling, and property testing in ordered graphs. In particular, we prove a new ordered analogue of the well-known result by Alon and Stav [RS\&A'08] on the furthest graph from a hereditary property; this is the first known result of this type in the ordered setting. Unlike the unordered regime, here the random graph model $G(n, p)$ with an ordering over the vertices is \emph{not} always asymptotically the furthest from the property for some $p$. However, using our ordered limit theory, we show that random graphs generated by a stochastic block model, where the blocks are consecutive in the vertex ordering, are (approximately) the furthest. Additionally, we describe an alternative analytic proof of the ordered graph removal lemma [Alon et al., FOCS'17].Comment: Added a new application: An Alon-Stav type result on the furthest ordered graph from a hereditary property; Fixed and extended proof sketch of the removal lemma applicatio

Hard Properties with (Very) Short PCPPs and Their Applications

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ?, we construct a property P^(?)? {0,1}^n satisfying the following: Any testing algorithm for P^(?) requires ?(n) many queries, and yet P^(?) has a constant query PCPP whose proof size is O(n?log^(?)n), where log^(?) denotes the ? times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n?polylog(n)). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ?, we construct a property that has a constant-query tester, but requires ?(n/log^(?)(n)) queries for every tolerant or erasure-resilient tester