23,493 research outputs found
Testing Hereditary Properties of Sequences
A hereditary property of a sequence is one that is preserved when restricting to subsequences. We show that there exist hereditary properties of sequences that cannot be tested with sublinear queries, resolving an open question posed by Newman et al. This proof relies crucially on an infinite alphabet, however; for finite alphabets, we observe that any hereditary property can be tested with a constant number of queries
On the Typical Structure of Graphs in a Monotone Property
Given a graph property , it is interesting to determine the
typical structure of graphs that satisfy . In this paper, we
consider monotone properties, that is, properties that are closed under taking
subgraphs. Using results from the theory of graph limits, we show that if
is a monotone property and is the largest integer for which
every -colorable graph satisfies , then almost every graph with
is close to being a balanced -partite graph.Comment: 5 page
Nondeterministic graph property testing
A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory of graph limits, we prove that
nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate,
describe new application
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
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
Graph properties, graph limits and entropy
We study the relation between the growth rate of a graph property and the
entropy of the graph limits that arise from graphs with that property. In
particular, for hereditary classes we obtain a new description of the colouring
number, which by well-known results describes the rate of growth.
We study also random graphs and their entropies. We show, for example, that
if a hereditary property has a unique limiting graphon with maximal entropy,
then a random graph with this property, selected uniformly at random from all
such graphs with a given order, converges to this maximizing graphon as the
order tends to infinity.Comment: 24 page
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