Showing papers by "Asaf Shapira published in 2018"

A generalized Turán problem and its applications

Abstract: Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f, there is a graph property whose 1-sided-error query complexity is precisely f(Θ(1/e)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/e). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/e). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose 1-sided-error query complexity is f(Θ(1/e)) while its 2-sided-error query complexity is only poly(1/e). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turan problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2l +1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cl) for all the remaining values of k and l.

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. Three examples of our results are the following: 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. Trevisan and Arora-Barak-Steurer have recently shown that given a graph G, one can remove only 1% of its edges to 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. Sudakov and the second author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(log⁡n)1+o(1). We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.

A Generalized Turan Problem and its Applications.

Abstract: Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f, there is a graph property whose 1-sided-error query complexity is precisely f(Θ(1/e)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/e). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/e). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose 1-sided-error query complexity is f(Θ(1/e)) while its 2-sided-error query complexity is only poly(1/e). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turan problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2l +1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cl) for all the remaining values of k and l.

Efficient Testing without Efficient Regularity

Abstract: The regularity lemma of Szemeredi turned out to be the most powerful tool for studying the testability of graph properties in the dense graph model. In fact, as we argue in this paper, this lemma can be used in order to prove (essentially) all the previous results in this area. More precisely, a barrier for obtaining an efficient testing algorithm for a graph property P was having an efficient regularity lemma for graphs satisfying P. The problem is that for many natural graph properties (e.g. triangle freeness) it is known that a graph can satisfy P and still only have regular partitions of tower-type size. This means that there was no viable path for obtaining reasonable bounds on the query complexity of testing such properties. In this paper we consider the property of being induced C_4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for testing this property. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox. We thus obtain the first example of an efficient testing algorithm that cannot be derived from an efficient version of the regularity lemma.

A Tight Bound for Hypergraph Regularity II

Abstract: The hypergraph regularity lemma -- the extension of Szemeredi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rodl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers' famous lower bound for graph regularity.

Two Erd\H{o}s--Hajnal-type Theorems in Hypergraphs

Abstract: The Erd\H{o}s--Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph $H$, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of $r$-uniform hypergraphs. A theorem of R\"odl asserts that if an $n$-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to $0$ or $1$) of size $\Omega(n)$. We prove that if a $3$-uniform hypergraph is non-universal then it contains an almost homogeneous set of size $\Omega(\log n)$. An example of R\"odl from 1986 shows that this bound is tight. Let $R_r(t)$ denote the size of the largest non-universal $r$-graph $G$ so that neither $G$ nor its complement contain a complete $r$-partite subgraph with parts of size $t$. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for $R_{r}(t)$ into a lower bound for $R_{r+1}(t)$. As an application of this lemma, we improve a bound of Conlon--Fox--Sudakov by showing that $R_3(t) \geq t^{\Omega(t)}$.

Two Erdős--Hajnal-type Theorems in Hypergraphs

Abstract: The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs. A theorem of Rodl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e. one with edge density either very close to 0 or 1) of size Ω ( n ) . We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size Ω ( log ⁡ n ) . An example of Rodl from 1986 shows that this bound is tight. Let R r ( t ) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erdős–Hajnal-type stepping-up lemma, showing how to transform a lower bound for R r ( t ) into a lower bound for R r + 1 ( t ) . As an application of this lemma, we improve a bound of Conlon–Fox–Sudakov by showing that R 3 ( t ) ≥ t Ω ( t ) .