Lior Gishboliner

Removal lemmas with polynomial bounds

We give new sufficient and necessary criteria guaranteeing that a hereditary graph property can be tested with a polynomial query complexity. Although both are simple combinatorial criteria, they imply almost all prior positive and negative results of this type, as well as many new ones. One striking application of our results is that every semi-algebraic graph property (e.g., being an interval graph, a unit-disc graph etc.) can be tested with a polynomial query complexity. This confirms a conjecture of Alon. The proofs combine probabilistic ideas together with a novel application of a conditional regularity lemma for matrices, due to Alon, Fischer and Newman.

A generalized Turán problem and its applications

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/ε)). 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/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). 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/ε)) while its 2-sided-error query complexity is only poly(1/ε). 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 Turán 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

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.

On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of

Deterministic vs non-deterministic graph property testing

G\sim G\left(n,\frac {1+\varepsilon}n\right)

Deterministic vs Non-deterministic Graph Property Testing.

in terms of

A Generalized Tur\'an Problem and its Applications

\varepsilon

Efficient Removal without Efficient Regularity

. We then apply this result to prove the following conjecture by Frieze and Pegden. For every

The Minrank of Random Graphs over Arbitrary Fields.

\varepsilon>0

The removal lemma for tournaments

there exists