Approximating the Distance to Monotonicity of Boolean Functions

Pallavoor, Ramesh Krishnan S.; Raskhodnikova, Sofya; Waingarten, Erik

doi:10.1137/1.9781611975994.123

Cited by 5 publications

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“…Erasure-resilient sublinear-time algorithms, in the context of testing properties of functions, were first investigated by Dixit et al [7], and further studied by Raskhodnikova et al [25], Pallavoor et al [22], and Ben-Eliezer et al [3].…”

Section: Related Workmentioning

confidence: 99%

Erasure-Resilient Sublinear-Time Graph Algorithms

Levi

Pallavoor

Raskhodnikova

et al. 2021

ACM Trans. Comput. Theory

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We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an “interpolation” between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. ‘06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms ‘08) and Eden et al. (ICALP ‘17). We conclude with a discussion of our model and open questions raised by our work.

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Section: Related Workmentioning

confidence: 99%

Erasure-Resilient Sublinear-Time Graph Algorithms

Levi

Pallavoor

Raskhodnikova

et al. 2021

ACM Trans. Comput. Theory

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“…Levi and Waingarten [48] showed that every algorithm approximating the distance to unateness within a constant factor requires Ω(d) queries and strengthened their lower bound to Ω(d 3/2 ) queries for nonadaptive algorithms. Subsequently, Pallavoor et al [50] proved that every nonadaptive algorithm approximating the distance to unateness within a d 1/2−k factor for k > 0 requires Sesh used to have hobbies and was an all around interesting person, but that was before his kids were born.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

“…Testing various properties of functions, including monotonicity (see, e. g., [38,52,33,34,47,36,53,35,41,1,42,5,16,12,19,22,17,11,23,24,21,27,26,45,8,9,32,49,29,7,13,25,14,50] and recent surveys [54,55,20]), the Lipschitz property [43,22,31,17,2], bounded-derivative properties [21], linearity [18,6,10,44,56], submodularity [51,58,60,…”

Section: Introductionmentioning

confidence: 99%

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Baleshzar¹,

Chakrabarty²,

Pallavoor³

et al. 2020

Theory of Comput.

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We study the problem of testing unateness of functions f : {0, 1} d → R. A function f : {0, 1} d → R is unate if for every coordinate i ∈ [d], the function is either nonincreasing in the i th coordinate or nondecreasing in the i th coordinate. We give an O((d/ε) • log(d/ε))-query nonadaptive tester and an O(d/ε)-query adaptive tester and show that both testers are optimal for a fixed distance parameter ε. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension d both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. (Concurrent work by Chen et al. (STOC'17) proved an Ω(d/ log 2 d) lower bound on the nonadaptive query complexity of testing unateness of Boolean functions.) We also generalize our results to obtain optimal unateness testers for functions f : [n] d → R.

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“…This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing. A preliminary version of this work appeared in the proceedings of ACM‐SIAM Symposium on Discrete Algorithms (SODA), 2020 [54].…”

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confidence: 99%

Approximating the distance to monotonicity of Boolean functions

Pallavoor

Raskhodnikova

Waingarten

2021

Random Struct Algorithms

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We design a nonadaptive algorithm that, given oracle access to a function f:{0,1} n→{0,1} which is α‐far from monotone, makes poly(n,1/α) queries and returns an estimate that, with high probability, is an Õ(n)‐approximation to the distance of f to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly(n,1/α)‐query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant κ>0, every nonadaptive n1/2−κ‐approximation algorithm for this problem requires 2nκ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure‐resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a k‐junta.

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Approximating the Distance to Monotonicity of Boolean Functions

Cited by 5 publications

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Erasure-Resilient Sublinear-Time Graph Algorithms

Erasure-Resilient Sublinear-Time Graph Algorithms

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Approximating the distance to monotonicity of Boolean functions

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