Showing papers by "Asaf Shapira published in 2015"

A unified framework for testing linear-invariant properties

Abstract: In the study of property testing, a particularly important role has been played by linear invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear-invariant properties, drawing on techniques from additive combinatorics and from graph theory.

Small complete minors above the extremal edge density

Abstract: A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the notion of high connectivity by the notion of vertex expansion. Another well known result in graph theory states that for every integer t there is a smallest real c(t), such that every n-vertex graph with c(t) n edges contains a K t -minor. Fiorini, Joret, Theis and Wood asked whether every n-vertex graph G that has at least (c(t)+?)n edges, must contain a K t -minor of order at most C(?) logn. We use our extension of Mader's theorem to prove that such a graph G must contain a K t -minor of order at most C(?) lognlognlogn. Known constructions of graphs with high girth show that this result is tight up to the log logn factor.

An Optimal Algorithm for Finding Frieze–Kannan Regular Partitions

Abstract: In this paper we prove that two local conditions involving the degrees and co-degrees in a graph can be used to determine whether a given vertex partition is Frieze{Kannan-regular. With a more rened version of these two local conditions we provide a deterministic algorithm that obtains a Frieze{Kanan-regular partition of any graph G in time O(jV (G)j 2 ).

Decomposing a Graph Into Expanding Subgraphs

Abstract: A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. These results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.

Constructing Near Spanning Trees with Few Local Inspections

Abstract: Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge $e$ in $G$ we would like to decide whether $e$ belongs to a connected subgraph $G'$ consisting of $(1+\epsilon)n$ edges (for a prespecified constant $\epsilon >0$), where the decision for different edges should be consistent with the same subgraph $G'$. Can this task be performed by inspecting only a {\em constant} number of edges in $G$? Our main results are: (1) We show that if every $t$-vertex subgraph of $G$ has expansion $1/(\log t)^{1+o(1)}$ then one can (deterministically) construct a sparse spanning subgraph $G'$ of $G$ using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm. (2) We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of $3$-regular graphs of high girth, in which every $t$-vertex subgraph has expansion $1/(\log t)^{1-o(1)}$.

Decomposing a graph into expanding subgraphs

Abstract: A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Two examples of our results are the following:• Motivated by the Unique Games Conjecture, Trevisan [FOCS '05] and Arora, Barak and Steurer [FOCS '10] showed that given a graph G, one can remove only 1% of G's edges and thus obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are (essentially) best possible even when one is allowed to remove 99% of G's edges. In particular, our results imply that the eigenspace enumeration approach of Arora-Barak-Steurer cannot give (even quasi-) polynomial time algorithms for unique games.• A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log n)1-o(1) such that not all graphs in the family have O(n) edges.The above results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.

A tournament approach to pattern avoiding matrices

Abstract: We consider the following Tur\'an-type problem: given a fixed tournament $H$, what is the least integer $t=t(n,H)$ so that adding $t$ edges to any $n$-vertex tournament, results in a digraph containing a copy of $H$. Similarly, what is the least integer $t=t(T_n,H)$ so that adding $t$ edges to the $n$-vertex transitive tournament, results in a digraph containing a copy of $H$. Besides proving several results on these problems, our main contributions are the following: (1) Pach and Tardos conjectured that if $M$ is an acyclic $0/1$ matrix, then any $n \times n$ matrix with $n(\log n)^{O(1)}$ entries equal to $1$ contains the pattern $M$. We show that this conjecture is equivalent to the assertion that $t(T_n,H)=n(\log n)^{O(1)}$ if and only if $H$ belongs to a certain (natural) family of tournaments. (2) We propose an approach for determining if $t(n,H)=n(\log n)^{O(1)}$. This approach combines expansion in sparse graphs, together with certain structural characterizations of $H$-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach-Tardos conjecture.