24 research outputs found
Local And Global Colorability of Graphs
It is shown that for any fixed and , the maximum possible
chromatic number of a graph on vertices in which every subgraph of radius
at most is colorable is (that is, up to a factor poly-logarithmic in ).
The proof is based on a careful analysis of the local and global colorability
of random graphs and implies, in particular, that a random -vertex graph
with the right edge probability has typically a chromatic number as above and
yet most balls of radius in it are -degenerate
Testing Local Properties of Arrays
We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of d-dimensional arrays f:[n]^d -> Sigma is k-local if it can be defined by a family of k x ... x k forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and submodularity are 2-local; convexity is (usually) 3-local; and many typical problems in computational biology and computer vision involve o(n)-local properties.
In this work, we present a generic approach to test all local properties of arrays over any finite (and not necessarily bounded size) alphabet. We show that any k-local property of d-dimensional arrays is testable by a simple canonical one-sided error non-adaptive epsilon-test, whose query complexity is O(epsilon^{-1}k log{(epsilon n)/k}) for d = 1 and O(c_d epsilon^{-1/d} k * n^{d-1}) for d > 1. The queries made by the canonical test constitute sphere-like structures of varying sizes, and are completely independent of the property and the alphabet Sigma. The query complexity is optimal for a wide range of parameters: For d=1, this matches the query complexity of many previously investigated local properties, while for d > 1 we design and analyze new constructions of k-local properties whose one-sided non-adaptive query complexity matches our upper bounds. For some previously studied properties, our method provides the first known sublinear upper bound on the query complexity
Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays
Understanding the local behaviour of structured multi-dimensional data is a
fundamental problem in various areas of computer science. As the amount of data
is often huge, it is desirable to obtain sublinear time algorithms, and
specifically property testers, to understand local properties of the data.
We focus on the natural local problem of testing pattern freeness: given a
large -dimensional array and a fixed -dimensional pattern over a
finite alphabet, we say that is -free if it does not contain a copy of
the forbidden pattern as a consecutive subarray. The distance of to
-freeness is the fraction of entries of that need to be modified to make
it -free. For any and any large enough pattern over
any alphabet, other than a very small set of exceptional patterns, we design a
tolerant tester that distinguishes between the case that the distance is at
least and the case that it is at most , with query
complexity and running time , where and
depend only on .
To analyze the testers we establish several combinatorial results, including
the following -dimensional modification lemma, which might be of independent
interest: for any large enough pattern over any alphabet (excluding a small
set of exceptional patterns for the binary case), and any array containing
a copy of , one can delete this copy by modifying one of its locations
without creating new -copies in .
Our results address an open question of Fischer and Newman, who asked whether
there exist efficient testers for properties related to tight substructures in
multi-dimensional structured data. They serve as a first step towards a general
understanding of local properties of multi-dimensional arrays, as any such
property can be characterized by a fixed family of forbidden patterns
Efficient Removal Lemmas for Matrices
The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing an (ordered) matrix removal lemma, which states the following: If a matrix is far from satisfying some hereditary property, then a large enough constant-size random submatrix of it does not satisfy the property with probability at least 9/10. Here being far from the property means that one needs to modify a constant fraction of the entries of the matrix to make it satisfy the property.
However, in the above general removal lemma, the required size of the random submatrix grows very fast as a function of the distance of the matrix from satisfying the property. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: If an epsilon-fraction of the entries of a binary matrix M can be covered by pairwise-disjoint copies of some (s x t) matrix A, then a delta-fraction of the (s x t)-submatrices of M are equal to A, where delta is polynomial in epsilon.
We generalize the work of Alon, Fischer and Newman [SICOMP\u2707] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas
Earthmover Resilience and Testing in Ordered Structures
One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures such as strings, images, and ordered graphs, the characterization problem seems very difficult in general.
In this paper, we identify a wide class of properties of ordered structures - the earthmover resilient (ER) properties - and show that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs. A property P is ER if, roughly speaking, slight changes in the order of the elements in an object satisfying P cannot make this object far from P. The class of ER properties includes, e.g., all unordered graph properties, many natural visual properties of images, such as convexity, and all hereditary properties of ordered graphs and images.
A special case of our results implies, building on a recent result of Alon and the authors, that the distance of a given image or ordered graph from any hereditary property can be estimated (with good probability) up to a constant additive error, using a constant number of queries
Limits of Ordered Graphs and their Applications
The emerging theory of graph limits exhibits an analytic perspective on
graphs, showing that many important concepts and tools in graph theory and its
applications can be described more naturally (and sometimes proved more easily)
in analytic language. We extend the theory of graph limits to the ordered
setting, presenting a limit object for dense vertex-ordered graphs, which we
call an \emph{orderon}. As a special case, this yields limit objects for
matrices whose rows and columns are ordered, and for dynamic graphs that expand
(via vertex insertions) over time. Along the way, we devise an ordered
locality-preserving variant of the cut distance between ordered graphs, showing
that two graphs are close with respect to this distance if and only if they are
similar in terms of their ordered subgraph frequencies. We show that the space
of orderons is compact with respect to this distance notion, which is key to a
successful analysis of combinatorial objects through their limits.
We derive several applications of the ordered limit theory in extremal
combinatorics, sampling, and property testing in ordered graphs. In particular,
we prove a new ordered analogue of the well-known result by Alon and Stav
[RS\&A'08] on the furthest graph from a hereditary property; this is the first
known result of this type in the ordered setting. Unlike the unordered regime,
here the random graph model with an ordering over the vertices is
\emph{not} always asymptotically the furthest from the property for some .
However, using our ordered limit theory, we show that random graphs generated
by a stochastic block model, where the blocks are consecutive in the vertex
ordering, are (approximately) the furthest. Additionally, we describe an
alternative analytic proof of the ordered graph removal lemma [Alon et al.,
FOCS'17].Comment: Added a new application: An Alon-Stav type result on the furthest
ordered graph from a hereditary property; Fixed and extended proof sketch of
the removal lemma applicatio
Hard Properties with (Very) Short PCPPs and Their Applications
We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ?, we construct a property P^(?)? {0,1}^n satisfying the following: Any testing algorithm for P^(?) requires ?(n) many queries, and yet P^(?) has a constant query PCPP whose proof size is O(n?log^(?)n), where log^(?) denotes the ? times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n?polylog(n)).
As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ?, we construct a property that has a constant-query tester, but requires ?(n/log^(?)(n)) queries for every tolerant or erasure-resilient tester