34 research outputs found
Testing Linear Inequalities of Subgraph Statistics
Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property ? and those that are far from satisfying it. Since these algorithms operate by inspecting a small randomly selected portion of the input, the most natural property one would like to be able to test is whether the input does not contain certain forbidden small substructures. In the setting of graphs, such a result was obtained by Alon et al., who proved that for any finite family of graphs ?, the property of being induced ?-free (i.e. not containing an induced copy of any F ? ?) is testable.
It is natural to ask if one can go one step further and prove that more elaborate properties involving induced subgraphs are also testable. One such generalization of the result of Alon et al. was formulated by Goldreich and Shinkar who conjectured that for any finite family of graphs ?, and any linear inequality involving the densities of the graphs F ? ? in the input graph, the property of satisfying this inequality can be tested in a certain restricted model of graph property testing. Our main result in this paper disproves this conjecture in the following strong form: some properties of this type are not testable even in the classical (i.e. unrestricted) model of graph property testing.
The proof deviates significantly from prior non-testability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a pseudo-random graph
Separations of Matroid Freeness Properties
Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest
Every locally characterized affine-invariant property is testable
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a
property of functions on F^n that is closed under taking affine transformations
of the domain. We prove that all affine-invariant property having local
characterizations are testable. In fact, we show a proximity-oblivious test for
any such property P, meaning that there is a test that, given an input function
f, makes a constant number of queries to f, always accepts if f satisfies P,
and rejects with positive probability if the distance between f and P is
nonzero. More generally, we show that any affine-invariant property that is
closed under taking restrictions to subspaces and has bounded complexity is
testable.
We also prove that any property that can be described as the property of
decomposing into a known structure of low-degree polynomials is locally
characterized and is, hence, testable. For example, whether a function is a
product of two degree-d polynomials, whether a function splits into a product
of d linear polynomials, and whether a function has low rank are all examples
of degree-structural properties and are therefore locally characterized.
Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low
characteristic fields that decomposes any polynomial with large Gowers norm
into a function of low-degree non-classical polynomials. We establish a new
equidistribution result for high rank non-classical polynomials that drives the
proofs of both the testability results and the local characterization of
degree-structural properties
Correlation Testing for Affine Invariant Properties on in the High Error Regime
Recently there has been much interest in Gowers uniformity norms from the
perspective of theoretical computer science. This is mainly due to the fact
that these norms provide a method for testing whether the maximum correlation
of a function with polynomials of
degree at most is non-negligible, while making only a constant number
of queries to the function. This is an instance of {\em correlation testing}.
In this framework, a fixed test is applied to a function, and the acceptance
probability of the test is dependent on the correlation of the function from
the property. This is an analog of {\em proximity oblivious testing}, a notion
coined by Goldreich and Ron, in the high error regime. In this work, we study
general properties which are affine invariant and which are correlation
testable using a constant number of queries. We show that any such property (as
long as the field size is not too small) can in fact be tested by Gowers
uniformity tests, and hence having correlation with the property is equivalent
to having correlation with degree polynomials for some fixed . We stress
that our result holds also for non-linear properties which are affine
invariant. This completely classifies affine invariant properties which are
correlation testable. The proof is based on higher-order Fourier analysis.
Another ingredient is a nontrivial extension of a graph theoretical theorem of
Erd\"os, Lov\'asz and Spencer to the context of additive number theory.Comment: 43 pages. A preliminary version of this work appeared in STOC' 201
Erasure-Resilient Property Testing
Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions, all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some cases, make the tester entirely useless.
In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain points to satisfy P.
We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box. However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size
An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs
We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by d, any property that can be tested with query complexity O_{?,d}(1) in the bidirectional model can be tested with n^{1-?_{?,d}(1)} queries in the unidirectional model. In particular, {if the proximity parameter ? approaches 0, then the query complexity of the transformed tester in the unidirectional model approaches n}. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation.
We prove that testing subgraph-freeness in which the subgraph contains k source components, requires ?(n^{1-1/k}) queries in the unidirectional model. This directly gives the first explicit properties that exhibit an O_{?,d}(1) vs ?(n^{1-f(?,d)}) separation of the query complexities between the bidirectional model and unidirectional model, where f(?,d) is a function that approaches 0 as ? approaches 0. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing k-star-freeness
Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness
An error-correcting code C subseteq F^n is called (q,epsilon)-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords x not in C with probability at least epsilon * delta(x,C), where delta(x,C) denotes the relative Hamming distance between the word x and the code C. The parameter q is called the query complexity and the parameter epsilon is called soundness.
Goldreich and Sudan (J.ACM 2006) asked about the existence of strong LTCs with constant query complexity, constant relative distance, constant soundness and inverse polylogarithmic rate. They also asked about the explicit constructions of these codes.
Strong LTCs with the required range of parameters were obtained recently in the works of Viderman (CCC 2013, FOCS 2013) based on the papers of Meir (SICOMP 2009) and Dinur (J.ACM 2007). However, the construction of these codes was probabilistic.
In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. Previously, such codes were proven to be weak LTCs. Using the results of Viderman (CCC 2013, FOCS 2013) we prove that such codes are, in fact, strong LTCs