14 research outputs found
A Hypergraph Dictatorship Test with Perfect Completeness
A hypergraph dictatorship test is first introduced by Samorodnitsky and
Trevisan and serves as a key component in their unique games based \PCP
construction. Such a test has oracle access to a collection of functions and
determines whether all the functions are the same dictatorship, or all their
low degree influences are Their test makes queries and has
amortized query complexity but has an inherent loss of
perfect completeness. In this paper we give an adaptive hypergraph dictatorship
test that achieves both perfect completeness and amortized query complexity
.Comment: Some minor correction
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
On sketching approximations for symmetric Boolean CSPs
A Boolean maximum constraint satisfaction problem, Max-CSP(), is specified
by a predicate . An -variable instance of
Max-CSP() consists of a list of constraints, each of which applies to
distinct literals drawn from the variables. For , Chou, Golovnev,
and Velusamy [CGV20, FOCS 2020] obtained explicit ratios characterizing the
-space streaming approximability of every predicate. For ,
Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general
dichotomy theorem for -space sketching algorithms: For every ,
there exists such that for every ,
Max-CSP() is -approximable by an -space
linear sketching algorithm, but -approximation sketching
algorithms require space.
In this work, we give closed-form expressions for the sketching approximation
ratios of multiple families of symmetric Boolean functions. Letting , we show that for odd ,
AND, and for even , AND. We also resolve the ratio for the "at-least--'s"
function for all even ; the "exactly--'s" function for odd
; and fifteen other functions. We stress here that for
general , according to [CGSV21], closed-form expressions for
need not have existed a priori.
Separately, for all threshold functions, we give optimal "bias-based"
approximation algorithms generalizing [CGV20] while simplifying [CGSV21].
Finally, we investigate the -space streaming lower bounds in [CGSV21],
and show that they are incomplete for AND.Comment: 27 pages; same results but significant changes in presentatio
Sublinear-Time Computation in the Presence of Online Erasures
We initiate the study of sublinear-time algorithms that access their input
via an online adversarial erasure oracle. After answering each query to the
input object, such an oracle can erase input values. Our goal is to
understand the complexity of basic computational tasks in extremely adversarial
situations, where the algorithm's access to data is blocked during the
execution of the algorithm in response to its actions. Specifically, we focus
on property testing in the model with online erasures. We show that two
fundamental properties of functions, linearity and quadraticity, can be tested
for constant with asymptotically the same complexity as in the standard
property testing model. For linearity testing, we prove tight bounds in terms
of , showing that the query complexity is . In contrast to
linearity and quadraticity, some other properties, including sortedness and the
Lipschitz property of sequences, cannot be tested at all, even for . Our
investigation leads to a deeper understanding of the structure of violations of
linearity and other widely studied properties. We also consider implications of
our results for algorithms that are resilient to online adversarial corruptions
instead of erasures
Property Testing with Online Adversaries
The online manipulation-resilient testing model, proposed by Kalemaj,
Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies
property testing in situations where access to the input degrades continuously
and adversarially. Specifically, after each query made by the tester is
answered, the adversary can intervene and either erase or corrupt data
points. In this work, we investigate a more nuanced version of the online model
in order to overcome old and new impossibility results for the original model.
We start by presenting an optimal tester for linearity and a lower bound for
low-degree testing of Boolean functions in the original model. We overcome the
lower bound by allowing batch queries, where the tester gets a group of queries
answered between manipulations of the data. Our batch size is small enough so
that function values for a single batch on their own give no information about
whether the function is of low degree. Finally, to overcome the impossibility
results of Kalemaj et al. for sortedness and the Lipschitz property of
sequences, we extend the model to include , i.e., adversaries that make
less than one erasure per query. For sortedness, we characterize the rate of
erasures for which online testing can be performed, exhibiting a sharp
transition from optimal query complexity to impossibility of testability (with
any number of queries). Our online tester works for a general class of local
properties of sequences. One feature of our results is that we get new (and in
some cases, simpler) optimal algorithms for several properties in the standard
property testing model.Comment: To be published in 15th Innovations in Theoretical Computer Science
(ITCS 2024
The Gowers norm in the testing of Boolean functions
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 65-68).A property tester is a fast, randomized algorithm that reads only a few entries of the input, and based on the values of these entries, it distinguishes whether the input has a certain property or is "different" from any input having this property. Furthermore, we say that a property tester has completeness c and soundness s if it accepts all inputs having the property with probability at least c and accepts "different" inputs with probability at most s + o(1). In this thesis we present two property testers for boolean functions on the boolean cube f0; 1gn. We summarize our contribution as follows. We present a new dictatorship test that determines whether the function is a dictator (of the form f(x) = xi for some coordinate i), or a function that is an "anti-dictator." Our test is "adaptive," makes q queries, has completeness 1, and soundness O(q3) 2??q. Previously, a dictatorship test that has soundness (q + 1) . 2-q is achieved by Samorodnitsky and Trevisan, but their test has completeness strictly less than 1. Furthermore, the previously best known dictatorship test from the PCP literature with completeness 1 has soundness ... . Our contribution lies in achieving perfect completeness and low sound- ness simultaneously. We consider properties of functions that are invariant under linear transformations of the boolean cube. Previous works, such as linearity testing and low-degree testing, have focused on linear properties.(cont.) The one exception is a test due to Green for "triangle freeness": a function f satisfies this property if f(x); f(y); f(x + y) do not all equal 1, for any pair x; y 2 f0; 1gn. We extend this test to a more systematic study and consider non-linear properties that are described by a single forbidden pattern. Specifically, let M denote an r by k matrix over f0; 1g. We say that a function f is M-free if there are no ~x = (x1,...,xk), where x1,...,xk 2 f0; 1gn such that f(x1),...,f(xk) = 1 and M~x = ~0. If M can be represented by an underlying graph, we can analyze a test that determines whether a function is M-free or \far" from one. Our test makes k queries, has completeness 1, and soundness bounded away from 1. The technique from our work leads to alternate proofs that some previously studied linear properties are testable, albeit with worse parameters. Our results, though quite different in terms of context, are connected by similar techniques. Our analysis of the algorithms relies on the machinery of the Gowers uniformity norm, a recent and powerful tool in additive combinatorics.by Victor Yen-Wen Chen.Ph.D
Symmetries in algebraic Property Testing
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D