Monotonicity testing over general poset domains
Eldar Fischer,Eric Lehman,Ilan Newman,Sofya Raskhodnikova,Ronitt Rubinfeld,Alex Samorodnitsky +5 more
- pp 474-483
TLDR
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.read more
Citations
More filters
Proceedings Article
Approximating the influence of monotone boolean functions in O(√n) query complexity
TL;DR: In this article, the authors presented a randomized algorithm that approximates the influence of a monotone Boolean function to within a multiplicative factor of (1 ± e) by performing O(√n log n/I[f] poly(1/e) queries.
Proceedings ArticleDOI
The Power and Limitations of Uniform Samples in Testing Properties of Figures
TL;DR: It is proved that convexity can be tested with \(O({Epsilon }^{-1})\) queries by testers that can make queries of their choice while uniform testers for this property require \(\varOmega ({\epsilon )^{-5/4}\) samples.
Proceedings ArticleDOI
Almost optimal super-constant-pass streaming lower bounds for reachability
TL;DR: In this article, the authors give an almost quadratic n2−o(1) lower bound on the space consumption of any o(√ logn)-pass streaming algorithm solving the (directed) s-t reachability problem.
Book ChapterDOI
An algebraic characterization of testable boolean CSPs
TL;DR: The hardness of testing Boolean CSPs is characterized in terms of the algebra generated by the relations used to form constraints, and it is conjecture that a CSP instance is testable in sublinear time if its Gaifman graph has bounded treewidth.
Journal ArticleDOI
Testing for forbidden order patterns in an array
TL;DR: In this paper, the authors studied the problem of testing the π-freeness of a sequence f : {1,..., n} → ℝ of length n contains a pattern π ∈ 𝔖k, k constant, be a (forbidden) pattern.
References
More filters
Book
The Probabilistic Method
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.
Book
Probability: Theory and Examples
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.
Journal ArticleDOI
A Separator Theorem for Planar Graphs
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.
Book ChapterDOI
A decomposition theorem for partially ordered sets
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.
A separator theorem for planar graphs
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.