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
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Posted Content
Domain Reduction for Monotonicity Testing: A $o(d)$ Tester for Boolean Functions in $d$-Dimensions.
TL;DR: In this paper, the authors give a monotonicity tester for Boolean functions over the continuous domain, where the distance to monotoneity is measured with respect to a product distribution over the domain.
Combinatorial Optimization On Massive Datasets: Streaming, Distributed, And Massively Parallel Computation
TL;DR: With the emergence of massive datasets across different application domains, there is a rapidly growing interest in solving various optimization problems over immense amounts of data, which raises the following fundamental question: how to design algorithms that are highly efficient in their resource usage.
Journal Article
Testing k-Monotonicity.
TL;DR: In this paper, the authors demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain for $k\geq 3$.
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Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs
Amit Levi,Erik Waingarten +1 more
TL;DR: It is shown that testing bipartiteness of $n$-nodes graphs using rejection sampling queries requires complexity $\widetilde{\Omega}(n^2)", which provides the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.
Parameterized Property Testing of Functions.
TL;DR: In this paper, the authors investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed, and find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small.
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