Arrow's Impossibility Theorem states that any constitution which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a Dictator has to be non-transitive. In this paper we study quantitative versions of Arrow theorem. Consider @math voters who vote independently at random, each following the uniform distribution over the 6 rankings of 3 alternatives. Arrow's theorem implies that any constitution which satisfies IIA and Unanimity and is not a dictator has a probability of at least @math for a non-transitive outcome. When @math is large, @math is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every @math , there exists a @math , which depends on @math only, such that for all @math , and all constitutions on 3 alternatives, if the constitution satisfies: The IIA condition. For every pair of alternatives @math , the probability that the constitution ranks @math above @math is at least @math . For every voter @math , the probability that the social choice function agrees with a dictatorship on @math at most @math . Then the probability of a non-transitive outcome is at least @math .
As noted in @cite_5 , there is an interesting connection between quantitative Arrow statements and the concept of testing introduced in @cite_12 @cite_16 which was studied and used extensively since. Roughly speaking a property of functions is testable if it is possible to perform a randomized test for the property such that if the probability that the function passes the test is close to @math , then the function has to be close to a function with the property (say in the hamming distance). In terms of testing, our result states that among all functions satisfying the IIA property, the Transitivity property is testable. Moreover, the natural test "works": i.e., in order to test for transitivity, one can pick a random input and check if the outcome is transitive.
@cite_5 @cite_16 @cite_12
1970630090 2018925011
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,l,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. We continue the study of combinatorial property testing, initiated by Goldreich, Goldwasser, and Ron in [J. ACM, 45 (1998), pp. 653--750]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language @math and an integer n there exists a randomized algorithm which always accepts a word w of length n if @math and rejects it with high probability if @math has to be modified in at least @math positions to create a word in L. The algorithm queries @math bits of w. This query complexity is shown to be optimal up to a factor polylogarithmic in @math . We also discuss the testability of more complex languages and show, in particular, that the query complexity required for testing context-free languages cannot be bounded by any function of @math . The problem of testing regular languages can be viewed as a part of a very general approach, seeking to probe testability of properties defined by logical means. A direct product function is a function of the form g(x_1, ldots, x_k)=(g_1(x_1), ldots, g_k(x_k)). We show that the direct product property is locally testable with two queries, that is, a canonical two-query test distinguishes between direct product functions and functions that are far from direct products with constant probability. This local testing question comes up naturally in the context of PCPs, where direct products play a prominent role for gap amplification. We consider the following natural two query test for a given function f:[N]^kto[M]^k begin quote bf Two query direct product test : Choose x, y that agree on a random set A of t coordinates and accept if f(x)_A=f(y)_A. End quote We provide a comprehensive analysis of this test for all parameters N, M, k, tle O(k) and success probability delta>0. Our main result is that if a given function f:[N]^kto[M]^k passes the test with probability delta ge 1-eps then there is a direct product function g such that Pr[f(x)=g(x)]ge 1-O(eps). This is the first result relating success in the above (or any) test to the fraction of the domain on which f is equal to a direct product function. This test has been analyzed in previous works for the case t ll k ll N, and results show closeness of f to a direct product under a less natural measure of "approximate agreement". In the small soundness regime, we prove that if the test above passes with probability delta ge exp(-k), then the function agrees with a direct product function on local parts of the domain. This extends the previous range of parameters of delta ge exp(-sqrt[3]k) to the entire meaningful range of delta>exp(-k). 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. 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 @math has an oblivious one-sided error tester if and only if @math 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. In most algorithmic applications which compare two distributions, information theoretic distances are more natural than standard l p norms. In this paper we design streaming and sublinear time property testing algorithms for entropy and various information theoretic distances. posed the problem of property testing with respect to the Jensen-Shannon distance. We present optimal algorithms for estimating bounded, symmetric f-divergences (including the Jensen-Shannon divergence and the Hellinger distance) between distributions in various property testing frameworks. Along the way, we close a (log n) H gap between the upper and lower bounds for estimating entropy H, yielding an optimal algorithm over all values of the entropy. In a data stream setting (sublinear space), we give the first algorithm for estimating the entropy of a distribution. Our algorithm runs in polylogarithmic space and yields an asymptotic constant factor approximation scheme. An integral part of the algorithm is an interesting use of an F 0 (the number of distinct elements in a set) estimation algorithm; we also provide other results along the space time approximation tradeoff curve.Our results have interesting structural implications that connect sublinear time and space constrained algorithms. The mediating model is the random order streaming model, which assumes the input is a random permutation of a multiset and was first considered by Munro and Paterson in 1980. We show that any property testing algorithm in the combined oracle model for calculating a permutation invariant functions can be simulated in the random order model in a single pass. This addresses a question raised by regarding the relationship between property testing and stream algorithms. Further, we give a polylog-space PTAS for estimating the entropy of a one pass random order stream. This bound cannot be achieved in the combined oracle (generalized property testing) model. The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions. In this article, we provide simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties, such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing over arbitrary ranges. We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show that this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties that relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best-known result. Given a function f : X rarr Sigma, its lscr-wise direct product is the function F = flscr : Xlscr rarr Sigmalscr defined by: F(x1,...,xlscr) = (f(x1),...,f(xlscr)). We are interested in the local testability of the direct product encoding (mapping f rarr flscr). Namely, given an arbitrary function F : Xlscr rarr Sigmalscr, we wish to determine how close it is to flscr for some f : X rarr Sigma, by making two random queries into F. In this work we analyze the case of low acceptance probability of the test. We show that even if the test passes with small probability, epsiv>0, already F must have a non-trivial structure and in particular must agree with some flscr on nearly epsiv of the domain. Moreover, we give a structural characterization of all functions F on which the test passes with probability epsiv. Our results can be viewed as a combinatorial analog of the low error dasialow degree testpsila, that is used in PCP constructions. We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of @math -dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties. The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family @math of forbidden vertex-ordered graphs, and any @math , there exist @math and @math so that any vertex-ordered graph which is @math -far from being @math -free contains at least @math copies of some @math (with the correct vertex order) where @math . The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for @math -partite graphs with "undesirable" edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph. Combinatorial property testing, initiated by Rubinfeld and Sudan [23] and formally defined by Goldreich, Goldwasser and Ron in [18], deals with the following relaxation of decision problems: Given a fixed property P and an input f, distinguish between the case that f satisfies P, and the case that no input that differs from f in less than some fixed fraction of the places satisfies P. An (e, q)-test for P is a randomized algorithm that queries at most q places of an input f and distinguishes with probability 2 3 between the case that f has the property and the case that at least an e-fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e. as a product order of total orders). The main result here presents an (e, poly(1 e))-test for every property of 0 1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the “monotonicity” property studied in [9,8,13], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less efficient) test for such properties of labeled grids with larger fixed size alphabets. All the above tests have in addition a 1-sided error probability. This class of properties is related to properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations. We also show that with one quantifier alternation, a certain property can be defined, for which no test with query complexity of O(n 1 4) (for a small enough fixed e) exists. The above results identify new classes of properties that are defined by means of restricted logics, and that are efficiently testable. They also lay out a platform that bridges some previous results. Tolerant testing is an emerging topic in the field of property testing, which was defined in [M. Parnas, D. Ron, and R. Rubinfeld, J. Comput. System Sci., 72 (2006), pp. 1012-1042] and has recently become a very active topic of research. In the general setting, there exist properties that are testable but are not tolerantly testable [E. Fischer and L. Fortnow, Proceedings of the @math th IEEE Conference on Computational Complexity, 2005, pp. 135-140]. On the other hand, we show here that in the setting of the dense graph model, all testable properties are not only tolerantly testable (which was already implicitly proved in [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451-476] and [O. Goldreich and L. Trevisan, Random Structures Algorithms, 23 (2003), pp. 23-57]), but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant. In the course of the proof we develop a framework for extending Szemeredi's regularity lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular, we construct a probabilistic algorithm that finds the parameters of a regular partition of an input graph using a constant number of queries, and an algorithm to find a regular partition of a graph using a @math circuit. This, in some ways, strengthens the results of [N. Alon, R. A. Duke, H. Lefmann, V. Rodl, and R. Yuster, J. Algorithms, 16 (1994), pp. 80-109]. We characterize the set of properties of Boolean-valued functions on a finite domain @math that are testable with a constant number of samples. Specifically, we show that a property @math is testable with a constant number of samples if and only if it is (essentially) a @math -part symmetric property for some constant @math , where a property is @math -part symmetric if there is a partition @math of @math such that whether @math satisfies the property is determined solely by the densities of @math on @math . We use this characterization to obtain a number of corollaries, namely: (i) A graph property @math is testable with a constant number of samples if and only if whether a graph @math satisfies @math is (essentially) determined by the edge density of @math . (ii) An affine-invariant property @math of functions @math is testable with a constant number of samples if and only if whether @math satisfies @math is (essentially) determined by the density of @math . (iii) For every constant @math , monotonicity of functions @math on the @math -dimensional hypergrid is testable with a constant number of samples. Convex and submodular functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well-known Monge matrices. A matrix @math is called a Monge matrix if for every @math and @math we have @math . If inequality holds in the opposite direction, then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix. In this work we present testing algorithms for the above properties. A testing algorithm for a predetermined property @math is given query access to an unknown function f and a distance parameter @math . The algorithm should accept f with high probability if it has the property @math and reject it with high probability if more than an @math -fraction of the function values should be modified so that f obtains the property. Our algorithm for testing whether a 1-dimensional function @math is convex (concave) has query complexity and running time of @math . Our algorithm for testing whether an n1 × n2 matrix V is a Monge (inverse Monge) matrix has query complexity and running time of @math . We initiate a thorough study of -- producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called testing model we emulate sequential tests for nearly all graph properties having @math -sided tests, while in the and models we obtain faster tests for triangle-freeness and bipartiteness respectively. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time as compared to their counterparts from the sequential querying model of traditional property testing. The simplest property testing algorithms allow a relatively smooth transitioning to the distributed model. For the more complex tasks we develop new machinery that is of independent interest. This includes a method for distributed maintenance of multiple random walks. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is far from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k-connected (for k > 1), cycle-free and Eulerian. Our algorithms work in time polynomial in 1 e, always accept the graph when it has the tested property, and reject with high probability if the graph is s-far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N-vertex d-degree graph for which more than edN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(√N) on the query complexity of testing algorithms for the bipartite and expander properties. We consider distribution-free property-testing of graph connectivity. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithm has an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, and testers were shown for very few properties. However, no distribution-free property testing algorithm was known for any graph property. We present the first distribution-free testing algorithms for one of the central properties in this area—graph connectivity (specifically, the problem is mainly interesting in the case of sparse graphs). We introduce three testing models for sparse graphs: A model for bounded-degree graphs, A model for graphs with a bound on the total number of edges (both models were already considered in the context of uniform distribution testing), and A model which is a combination of the two previous testing models; i.e., bounded-degree graphs with a bound on the total number of edges. We prove that connectivity can be tested in each of these testing models, in a distribution-free manner, using a number of queries that is independent of the size of the graph. This is done by providing a new analysis to previously known connectivity testers (from “standard”, uniform distribution property-testing) and by introducing some new testers. Data dimensionality is a crucial issue in a variety of settings, where it is desirable to determine whether a data set given in a high-dimensional space adheres to a low-dimensional structure. We study this problem in the framework of property testing: Given a query access to a data set S, we wish to determine whether S is low-dimensional, or whether it should be modified significantly in order to have the property. Allowing a constant probability of error, we aim at algorithms whose complexity does not depend on the size of S.We present algorithms for testing the low-dimensionality of a set of vectors and for testing whether a matrix is of low rank. We then address low-dimensionality in metric spaces. For vectors in the metric space l 1 , we show that low-dimensionality is not testable. For l 2 , we show that a data set can be tested for having a low-dimensional structure, but that the property of approximately having such a structure is not testable. In property testing, we are given oracle access to a function f, and we wish to test if the function satisfies a given property P, or it is e-far from having that property. In a more general setting, the domain on which the function is defined is equipped with a probability distribution, which assigns different weight to different elements in the domain. This paper relates the complexity of testing the monotonicity of a function over the d-dimensional cube to the Shannon entropy of the underlying distribution. We provide an improved upper bound on the query complexity of the property tester. We introduce the notion of a canonical tester for a class of properties on distributions, that is, a tester strong and general enough that “a distribution property in the class is testable if and only if the canonical tester tests it.” We construct a canonical tester for the class of properties of one or two distributions that are symmetric and satisfy a certain weak continuity condition. Analyzing the performance of the canonical tester on specific properties resolves two open problems, establishing lower bounds that match known upper bounds: we show that distinguishing between entropy @math on distributions over @math requires @math samples, and distinguishing whether a pair of distributions has statistical distance @math requires @math samples. Our techniques also resolve a conjecture about a property that our canonical tester does not apply to: distinguishing identical distributions from those with statistical distance @math requires @math samples. We consider monotonicity testing of functions f:[n]d→ 0,1 , in the property testing framework of Rubinfeld and Sudan [23] and Goldreich, Goldwasser and Ron [14]. Specifically, we consider the framework of distribution-free property testing, where the distance between functions is measured with respect to a fixed but unknown distribution D on the domain and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. We show that, though in the uniform distribution case, testing of boolean functions defined over the boolean hypercube can be done using query complexity that is polynomial in @math and in the dimension d, in the distribution-free setting such testing requires a number of queries that is exponential in d. Therefore, in the high-dimensional case (in oppose to the low-dimensional case), the gap between the query complexity for the uniform and the distribution-free settings is exponential.