614 research outputs found
A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems
Let be a property of function for
a fixed prime . An algorithm is called a tester for if, given
a query access to the input function , with high probability, it accepts
when satisfies and rejects when is "far" from satisfying
. In this paper, we give a characterization of affine-invariant
properties that are (two-sided error) testable with a constant number of
queries. The characterization is stated in terms of decomposition theorems,
which roughly claim that any function can be decomposed into a structured part
that is a function of a constant number of polynomials, and a pseudo-random
part whose Gowers norm is small. We first give an algorithm that tests whether
the structured part of the input function has a specific form. Then we show
that an affine-invariant property is testable with a constant number of queries
if and only if it can be reduced to the problem of testing whether the
structured part of the input function is close to one of a constant number of
candidates.Comment: 27 pages, appearing in STOC 2014. arXiv admin note: text overlap with
arXiv:1306.0649, arXiv:1212.3849 by other author
Maximum flow is approximable by deterministic constant-time algorithm in sparse networks
We show a deterministic constant-time parallel algorithm for finding an
almost maximum flow in multisource-multitarget networks with bounded degrees
and bounded edge capacities. As a consequence, we show that the value of the
maximum flow over the number of nodes is a testable parameter on these
networks.Comment: 8 page
Testing Low Complexity Affine-Invariant Properties
Invariance with respect to linear or affine transformations of the domain is
arguably the most common symmetry exhibited by natural algebraic properties. In
this work, we show that any low complexity affine-invariant property of
multivariate functions over finite fields is testable with a constant number of
queries. This immediately reproves, for instance, that the Reed-Muller code
over F_p of degree d < p is testable, with an argument that uses no detailed
algebraic information about polynomials except that low degree is preserved by
composition with affine maps.
The complexity of an affine-invariant property P refers to the maximum
complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of
linear forms used to characterize P. A more precise statement of our main
result is that for any fixed prime p >=2 and fixed integer R >= 2, any
affine-invariant property P of functions f: F_p^n -> [R] is testable, assuming
the complexity of the property is less than p. Our proof involves developing
analogs of graph-theoretic techniques in an algebraic setting, using tools from
higher-order Fourier analysis.Comment: 38 pages, appears in SODA '1
A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing
Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for
semi-algebraic -uniform hypergraphs of bounded complexity, showing that for
each the vertex set can be equitably partitioned into a bounded
number of parts (in terms of and the complexity) so that all but an
-fraction of the -tuples of parts are homogeneous. We prove that
the number of parts can be taken to be polynomial in . Our improved
regularity lemma can be applied to geometric problems and to the following
general question on property testing: is it possible to decide, with query
complexity polynomial in the reciprocal of the approximation parameter, whether
a hypergraph has a given hereditary property? We give an affirmative answer for
testing typical hereditary properties for semi-algebraic hypergraphs of bounded
complexity
Relating two property testing models for bounded degree directed graphs
We study property testing algorithms in directed graphs (digraphs) with maximum indegree and maximum outdegree upper bounded by d. For directed graphs with bounded degree, there are two different models in property testing introduced by Bender and Ron (2002). In the bidirectional model, one can access both incoming and outgoing edges while in the unidirectional model one can only access outgoing edges. In our paper we provide a new relation between the two models: we prove that if a property can be tested with constant query complexity in the bidirectional model, then it can be tested with sublinear query complexity in the unidirectional model. A corollary of this result is that in the unidirectional model (the model allowing only queries to the outgoing neighbors), every property in hyperfinite digraphs is testable with sublinear query complexity
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
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
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