19 research outputs found
Random local algorithms
Consider the problem when we want to construct some structure on a bounded
degree graph, e.g. an almost maximum matching, and we want to decide about each
edge depending only on its constant radius neighbourhood. We show that the
information about the local statistics of the graph does not help here. Namely,
if there exists a random local algorithm which can use any local statistics
about the graph, and produces an almost optimal structure, then the same can be
achieved by a random local algorithm using no statistics.Comment: 9 page
Testing bounded arboricity
In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are -close to having
arboricity and graphs that -far from having
arboricity , where is an absolute small constant. The query
complexity and running time of the algorithm are
where denotes
the number of vertices and denotes the number of edges. In terms of the
dependence on and this bound is optimal up to poly-logarithmic factors
since queries are necessary (and .
We leave it as an open question whether the dependence on can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling
Property Testing via Set-Theoretic Operations
Given two testable properties and , under
what conditions are the union, intersection or set-difference of these two
properties also testable? We initiate a systematic study of these basic
set-theoretic operations in the context of property testing. As an application,
we give a conceptually different proof that linearity is testable, albeit with
much worse query complexity. Furthermore, for the problem of testing
disjunction of linear functions, which was previously known to be one-sided
testable with a super-polynomial query complexity, we give an improved analysis
and show it has query complexity O(1/\eps^2), where \eps is the distance
parameter.Comment: Appears in ICS 201
Testing probability distributions underlying aggregated data
In this paper, we analyze and study a hybrid model for testing and learning
probability distributions. Here, in addition to samples, the testing algorithm
is provided with one of two different types of oracles to the unknown
distribution over . More precisely, we define both the dual and
cumulative dual access models, in which the algorithm can both sample from
and respectively, for any ,
- query the probability mass (query access); or
- get the total mass of , i.e. (cumulative
access)
These two models, by generalizing the previously studied sampling and query
oracle models, allow us to bypass the strong lower bounds established for a
number of problems in these settings, while capturing several interesting
aspects of these problems -- and providing new insight on the limitations of
the models. Finally, we show that while the testing algorithms can be in most
cases strictly more efficient, some tasks remain hard even with this additional
power
Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs
In this paper, we consider lower bounds on the query complexity for testing
CSPs in the bounded-degree model.
First, for any ``symmetric'' predicate except \equ
where , we show that every (randomized) algorithm that distinguishes
satisfiable instances of CSP(P) from instances -far
from satisfiability requires queries where is the
number of variables and is a constant that depends on and
. This breaks a natural lower bound , which is
obtained by the birthday paradox. We also show that every one-sided error
tester requires queries for such . These results are hereditary
in the sense that the same results hold for any predicate such that
. For EQU, we give a one-sided error tester
whose query complexity is . Also, for 2-XOR (or,
equivalently E2LIN2), we show an lower bound for
distinguishing instances between -close to and -far
from satisfiability.
Next, for the general k-CSP over the binary domain, we show that every
algorithm that distinguishes satisfiable instances from instances
-far from satisfiability requires queries. The
matching NP-hardness is not known, even assuming the Unique Games Conjecture or
the -to- Conjecture. As a corollary, for Maximum Independent Set on
graphs with vertices and a degree bound , we show that every
approximation algorithm within a factor d/\poly\log d and an additive error
of requires queries. Previously, only super-constant
lower bounds were known
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