Hypergraph 2-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen 2-colorable 3-uniform hypergraph. Lee, Molla, and Nagle recently extended this to k-uniform hypergraphs for all k ≥ 3. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants. Our first result in this paper is a new simple and elementary deterministic 2-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only O(n). Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a 2-coloring of an average 2-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex v, assigns color red/blue to v while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal 2-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time O(1).

A partition property of order k asks if a graph can be partitioned into k vertex sets of prescribed sizes so that the densities between any pair of sets falls within a prescribed range. This family of properties has been extensively studied in various areas of research ranging from theoretical computer science to statistical physics. Our main result is that every partition property of order k is testable with query complexity poly(k/ɛ). We thus obtain an exponential improvement (in k) over the (1/ɛ)O(k) bound obtained by Goldreich, Goldwasser and Ron in their seminal FOCS 1996 paper. We further prove that our bound is tight in the sense that it cannot be made sub-polynomial in either k or ɛ.

Abstract. Graph modification problems ask for the minimal number of vertex/edge additions/deletions needed to make a graph satisfy some predetermined property. A (meta-)problem of this type, which was raised by Yannakakis in 1981, asks to determine for which properties [Formula: see text] it is NP-hard to compute the smallest number of edge deletions needed to make a graph satisfy [Formula: see text]. Despite being extensively studied in the past 40 years, this problem is still wide open. In fact, it is open even when [Formula: see text] is the property of being [Formula: see text]-free, for some fixed graph [Formula: see text]. In this case we use [Formula: see text] to denote the smallest number of edge deletions needed to turn [Formula: see text] into an [Formula: see text]-free graph. Alon, Shapira, and Sudakov proved that if [Formula: see text] is not bipartite, then computing [Formula: see text] is NP-hard. They left open the problem of classifying the bipartite graphs [Formula: see text] for which computing [Formula: see text] is NP-hard. In this paper we resolve this problem when [Formula: see text] is a forest, showing that computing [Formula: see text] is polynomial-time solvable if [Formula: see text] is a star forest and NP-hard otherwise. Our main innovation in this work lies in introducing a new graph-theoretic approach for Yannakakis’s problem, which differs significantly from all prior works on this subject. In particular, we prove new results concerning an old and famous conjecture of Erdős and Sós, which are of independent interest.

A graph G on n vertices is ε-far from property P if one should add/delete at least ε n² edges to turn G into a graph satisfying P. A distance estimator for P is an algorithm that given G and α, ε > 0 distinguishes between the case that G is (α-ε)-close to 𝒫 and the case that G is α-far from 𝒫. If P has a distance estimator whose query complexity depends only on ε, then P is said to be estimable. Every estimable property is clearly also testable, since testing corresponds to estimating with α = ε. A central result in the area of property testing is the Fischer-Newman theorem, stating that an inverse statement also holds, that is, that every testable property is in fact estimable. The proof of Fischer and Newmann was highly ineffective, since it incurred a tower-type loss when transforming a testing algorithm for P into a distance estimator. This raised the natural problem, studied recently by Fiat-Ron and by Hoppen-Kohayakawa-Lang-Lefmann-Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results. - We show that if P is hereditary, then one can turn a tester for P into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et. al., who obtained a transformation with a double exponential loss. - We show that for every P, one can turn a testing algorithm for P into a distance estimator with a double exponential loss. This improves over the transformation of Fischer-Newman that incurred a tower-type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer-Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze-Kannan Weak Regular partitions that are of independent interest.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C 2k and C 2k+1 in n -vertex graphs of bounded degeneracy in time Õ( n d k ), where the fastest known algorithm for detecting directed copies of C k in general m -edge digraphs runs in time Õ( m d k ). Conversely, one can transform any O(n b k ) algorithm for computing the number of homomorphic copies of C 2k or of C 2k+1 in n -vertex graphs of bounded degeneracy, into an Õ( m b k ) time algorithm for detecting directed copies of C k in general m -edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C k -homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k -cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time , where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time . Conversely, one can transform any algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

We consider the problem of counting the number of copies of a fixed graph H within an input graph G . This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy . This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from ’85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices and further proved that for every \( k \gt 5 \) there is a k -vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H . Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera–Pashanasangi–Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.

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.

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.

We give new sufficient and necessary criteria guaranteeing that

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 O5] 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.

The Frieze–Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. R. Williams recently asked if one can construct a partition satisfying the conditions of the Frieze–Kannan regularity lemma in deterministic subcubic time. We resolve this problem by designing an $\tilde O(n^{\omega})$ time algorithm for constructing such a partition, where $\omega < 2.376$ is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze–Kannan regularity lemma.

We consider the performance of two classic approximation algorithms which work by scanning the input and greedily constructing a solution. We investigate whether running these algorithms on a random permutation of the input can increase their performance ratio. We obtain the following results: 1. Johnson's approximation algorithm for MAX-SAT is one of the first approximation algorithms to be rigorously analyzed. It has been shown that the performance ratio of this algorithm is 2/3. We show that when executed on a random permutation of the variables, the performance ratio of this algorithm is improved to 2/3 + c for some c > 0 This resolves an open problem of Chen, Friesen and Zhang [JCSS 1999]. (See also the paper by Poloczek and Schnitger in these proceedings for related results on this algorithm and its variants). 2. Motivated by the above improvement, we consider the performance of the greedy algorithm for MAX-CUT whose performance ratio is 1/2. Our hope was that running the greedy algorithm on a random permutation of the vertices would result in a 1/2 + c approximation algorithm. However, it turns out that in this case the performance of the algorithm remains 1/2. This resolves an open problem of Mathieu and Schudy [SODA 2008].

Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input. We discuss the types of answers that one can hope to achieve in this setting.

We show that, for every 0 ≤ p ≤ 1, there is an O ( n 2.575− p /(7.4−2.3 p ) )-time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u , v in the graph to within an additive error δ p ( u , v ), where δ( u , v ) is the exact length of the shortest path between u and v . This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1. Previously the only way to “beat” the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [2002] that approximates the shortest path distances within a multiplicative error of (1 + ϵ). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ϵ) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when ϵ ≪ 1 and the graph has small integer weights.

There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F2-linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan. We obtain the following results: 1. We show that every linear-invariant property that can be characterized by forbidding induced solutions to a (possibly infinite) set of linear equations can be tested with one-sided error. 2. We show that every linear-invariant property that can be tested with one-sided error can be characterized by forbidding induced solutions to a (possibly infinite) set of systems of linear equations. We conjecture that our result from item (1) can be extended to cover systems of linear equations. We further show that the validity of this conjecture would have the following implications: 1. It would imply that every linear-invariant property that is closed under restrictions to linear subspaces is testable with one-sided error. Such a result would unify several previous results on testing Boolean functions, such as the testability of low-degree polynomials and of Fourier dimensionality. 2. It would imply that a linear-invariant property P is testable with one-sided error if and only if P is closed under restrictions to linear subspaces, thus resolving Sudan's problem.

Szemerédi's regularity lemma is a cornerstone result in extremal combinatorics. It (roughly) asserts that any dense graph is composed of a finite number of pseudorandom graphs. The regularity lemma has found many applications in theoretical computer science, and thus a lot of attention was given to designing algorithmic versions of this lemma. Our main results in this paper are the following: (i) We introduce a new approach to the problem of constructing regular partitions of graphs, which results in a surprisingly simple $O(n)$ time algorithmic version of the regularity lemma, thus improving over the previous $O(n^2)$ time algorithms. Furthermore, unlike all the previous approaches for this problem (see [N. Alon and A. Naor, SIAM J. Comput., 35 (2006), pp. 787–803], [R. A. Duke, H. Lefmann, and V. Rödl, SIAM J. Comput., 24 (1995), pp. 598–620], [A. Frieze and R. Kannan, Electron. J. Combin., 6 (1999), article 17], [A. Frieze and R. Kannan, “The regularity lemma and approximation schemes for dense problems,” in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12–20], and [Y. Kohayakawa, V. Rödl, and L. Thoma, SIAM J. Comput., 32 (2003), pp. 1210–1235]), which only guaranteed to find tower-size partitions, our algorithm will find a small regular partition, if one exists in the graph. (ii) For any constant $r\geq3$ we give an $O(n)$ time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous $O(n^{2r-1})$ time (deterministic) algorithms [A. Czygrinow and V. Rödl, SIAM J. Comput., 30 (2000), pp. 1041–1066], [A. Frieze and R. Kannan, “The regularity lemma and approximation schemes for dense problems,” in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12–20]. These two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here: Given a (directed) hypergraph with bounded edge arities, a set of constraints on the set sizes and densities of a possible partition of its vertex set, and an approximation parameter, we provide in $O(n)$ time a partition approximating the constraints if a partition satisfying them exists. We can also test in $O(1)$ time for the existence of such a partition given the approximation parameter. This algorithm extends the result of Goldreich, Goldwasser, and Ron for graph partition problems [O. Goldreich, S. Goldwasser, and D. Ron, J. ACM, 45 (1998), pp. 653–750] and encompasses more recent hypergraph-related results such as the maximal constraint satisfaction approximation of [G. Andersson and L. Engebretsen, Random Structures Algorithms, 21 (2002), pp. 14–32].

A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green’s conjecture by showing that every set of linear equations (even non-homogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [4] related to algorithms for testing properties of boolean functions. 1 Background on removal lemmas The (triangle) removal lemma of Ruzsa and Szemerédi [18], which is by now a cornerstone result in combinatorics, states that a graph on n vertices that contains only o(n 3) triangles can be made triangle free by the removal of only o(n 2) edges. Or in other words, if a graph has asymptomatically few triangles then it is asymptotically close to being triangle free. While the lemma was proved

A common thread in all of the recent results concerning the testing of dense graphs is the use of Szemerédi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property ${\cal P}$ can be tested with a constant number of queries if and only if testing ${\cal P}$ can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised by Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653–750] in the paper that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable.

We study graph properties that are testable for bounded-degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is well known that in the bounded-degree graph model (where two graphs are considered “far” if they differ in $\varepsilon n$ edges for a positive constant $\varepsilon$), many graph properties cannot be tested even with a constant or even with a polylogarithmic number of queries. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. Specifically, we show that every hereditary graph property is testable with a constant number of queries provided that every sufficiently large induced subgraph of the input graph has poor expansion. This result implies that, for example, any hereditary property (e.g., k-colorability, H-freeness, etc.) is testable in the bounded-degree graph model for planar graphs, graphs with bounded genus, interval graphs, etc. No such results have been known before, and prior to our work, very few graph properties have been known to be testable with a constant number of queries for general graph classes in the bounded-degree graph model.

The regularity lemma of Szemerédi gives a concise approximate description of a graph via a so-called regular partition of its vertex set. In this paper we address the following problem: Can a graph have two “distinct” regular partitions? It turns out that (as observed by several researchers) for the standard notion of a regular partition, one can construct a graph that has very distinct regular partitions. On the other hand, we show that for the stronger notion of a regular partition that has been recently studied, all such regular partitions of the same graph must be very “similar.” En route, we also give a short argument for deriving a recent variant of the regularity lemma obtained independently by Rödl and Schacht and by Lovász and Szegedy from a previously known variant of the regularity lemma due to Alon et al. in 2000. The proof also provides a deterministic polynomial time algorithm for finding such partitions.

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property testing. Our main result in this paper is a solution of an important special case of this general problem: Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property ${\cal P}$ has an oblivious one-sided error tester if and only if ${\cal P}$ is semihereditary. We stress that any “natural” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the natural graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. More importantly, as a special case of our main result, we infer that some of the most well-studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well-known graph properties of being perfect, chordal, interval, comparability, permutation, and more. None of these properties was previously known to be testable.

A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on $\epsilon$. This result unifies several previous results in the area of property testing and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi's regularity lemma. The main ideas behind this application may be useful in characterizing all testable graph properties and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph theory, is that for any monotone graph property ${\cal P}$, any graph that is $\epsilon$-far from satisfying ${\cal P}$ contains a subgraph of size depending on $\epsilon$ only, which does not satisfy ${\cal P}$. Finally, we prove the following compactness statement: If a graph G is $\epsilon$-far from satisfying a (possibly infinite) set of monotone graph properties ${\cal P}$, then it is at least $\delta_{{\cal P}}(\epsilon)$-far from satisfying one of the properties.

We show that any partition-problem of hypergraphs has an O(n) time approximate partitioning algorithm and an efficient property tester. This extends the results of Goldreich, Goldwasser and Ron who obtained similar algorithms for the special case of graph partition problems in their seminal paper (1998). The partitioning algorithm is used to obtain the following results: ldr We derive a surprisingly simple O(n) time algorithmic version of Szemeredi's regularity lemma. Unlike all the previous approaches for this problem which only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists; ldr For any r ges 3, we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n2r-1) time (deterministic) algorithms. The property testing algorithm is used to unify several previous results, and to obtain the partition densities for the above problems (rather than the partitions themselves) using only poly(1/isin) queries and constant running time.

A common thread in recent results concerning the testing of dense graphs is the use of Szemerédi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron [25] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable.

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property-testing. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious one-sided error tester, if and only if P is (semi) hereditary. We stress that any "natural" property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the "natural" graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of Goldreich et al., [1998] about testing k-colorability, the characterization of Goldreich and Trevisan [2001] of the graph-partition problems that are testable with 1-sided error, the induced vertex colorability properties of Alon et al., [2000], the induced edge colorability properties of Fischer [2001], a transformation from 2-sided to 1-sided error testing [Goldreich and Trevisan, 2001], as well as a recent result about testing monotone graph properties [Alon and Shapira, 2005]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well known graph properties of being perfect, chordal, interval, comparability and more. None of these properties was previously known to be testable.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

A graph property is called monotone if it is closed under taking (not necessarily induced) subgraphs (or, equivalently, if it is closed under removal of edges and vertices). Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on ε. This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi's Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing.As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is ε -far from satisfying P, contains a subgraph of size depending on ε only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is ε-far from satisfying a (possibly infinite) set of graph properties P, then it is at least δ P ε-far from satisfying one of the properties.

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least ε n2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(ε,H) nh copies of H. This is proved by establishing a directed version of Szemeredi's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of ε only for testing the property PH of being H-free.