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Proceedings ArticleDOI

Monotonicity testing over general poset domains

TL;DR: It is shown that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory.
Abstract: The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are 'far' from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, primarily in the context of the monotonicity of a sequence of integers, or the monotonicity of a function over the n-dimensional hypercube {1,…,m}n. These works resulted in monotonicity testers whose query complexity is at most polylogarithmic in the size of the domain.We show that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory. These problems include: testing that a Boolean assignment of variables is close to an assignment that satisfies a specific 2-CNF formula, testing that a set of vertices is close to one that is a vertex cover of a specific graph, and testing that a set of vertices is close to a clique.We then investigate the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders. We give algorithms and lower bounds for the general problem, as well as for some interesting special cases. In proving a general lower bound, we construct graphs with combinatorial properties that may be of independent interest.
Citations
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Journal Article
TL;DR: In this paper, the authors consider the question of determining whether a function f has property P or is e-far from any function with property P. In some cases, it is also allowed to query f on instances of its choice.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

870 citations

Book
01 Nov 2017
TL;DR: In this article, a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions are presented.
Abstract: Property testing is concerned with the design of super-fast algorithms for the structural analysis of large quantities of data. The aim is to unveil global features of the data, such as determining whether the data has a particular property or estimating global parameters. Remarkably, it is possible for decisions to be made by accessing only a small portion of the data. Property testing focuses on properties and parameters that go beyond simple statistics. This book provides an extensive and authoritative introduction to property testing. It provides a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions.

343 citations

Book
Dana Ron1
27 Jan 2010
TL;DR: This monograph surveys results in property testing, where the emphasis is on common analysis and algorithmic techniques.
Abstract: Property testing algorithms are "ultra"-efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concerns properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this monograph we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: The self-correcting approach, which was mainly applied in the study of property testing of algebraic properties; The enforce-and-test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the dense-graphs model), as well as in other contexts; Szemeredi's Regularity Lemma, which plays a very important role in the analysis of algorithms for testing graph properties (in the dense-graphs model); The approach of Testing by implicit learning, which implies efficient testability of membership in many functions classes; and Algorithmic techniques for testing properties of sparse graphs, which include local search and random walks.

231 citations

Proceedings ArticleDOI
21 May 2006
TL;DR: One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.
Abstract: A common thread in recent results concerning the testing of dense graphs is the use of Szemeredi'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 Szemeredi-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 Szemeredi-partitions. This means that in some sense, testing for Szemeredi-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.

209 citations

Journal Article
TL;DR: This paper formalizes the notions of tolerant property testing and distance approximation and discusses the relationship between the two tasks, as well as their relationship to standard property testing.
Abstract: In this paper we study a generalization of standard property testing where the algorithms are required to be more tolerant with respect to objects that do not have, but are close to having, the property. Specifically, a tolerant property testing algorithm is required to accept objects that are e1 -close to having a given property P and reject objects that are e2-far from having P, for some parameters 0 ≤ e1 < e2 1. Another related natural extension of standard property testing that we study, is distance approximation. Here the algorithm should output an estimate e of the distance of the object to P, where this estimate is sufficiently close to the true distance of the object to P. We first formalize the notions of tolerant property testing and distance approximation and discuss the relationship between the two tasks, as well as their relationship to standard property testing. We then apply these new notions to the study of two problems: tolerant testing of clustering and distance approximation for monotonicity. We present and analyze algorithms whose query complexity is either polylogarithmic or independent of the size of the input.

163 citations

References
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Book
01 Jan 1991
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Abstract: The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

6,594 citations

Book
Rick Durrett1
01 Jan 1990
TL;DR: In this paper, a comprehensive introduction to probability theory covering laws of large numbers, central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion is presented.
Abstract: This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

5,168 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.

1,312 citations

Book ChapterDOI
TL;DR: In this article, a partially ordered set P is considered and two elements a and b of P are camparable if either a ≧ b or b ≧ a. If b and a are non-comparable, then they are independent.
Abstract: Let P be a partially ordered set. Two elements a and b of P are camparable if either a ≧ b or b ≧ a. Otherwise a and b are non-comparable. A subset S of P is independent if every two distinct elements of S are non-comparable. S is dependent if it contains two distinct elements which are comparable. A subset C of P is a chain if every two of its elements are comparable.

1,274 citations


"Monotonicity testing over general p..." refers background in this paper

  • ...By Dilworth’s theorem [7],+ + is equal to the minimum number of disjoint chains that cover ....

    [...]

01 Oct 1977
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A,B,C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A,B,C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2\sqrt{2}\sqrt{2}$ vertices. We exhibit an algorithm which finds such a partition A,B,C in O(n) time.

1,264 citations


"Monotonicity testing over general p..." refers background in this paper

  • ..., forests are -separable, bounded tree-width graphs have bounded separators and planar graphs are -separable [20]....

    [...]