52 research outputs found
Every Minor-Closed Property of Sparse Graphs is Testable
Suppose is a graph with degrees bounded by , and one needs to remove
more than of its edges in order to make it planar. We show that in
this case the statistics of local neighborhoods around vertices of is far
from the statistics of local neighborhoods around vertices of any planar graph
with the same degree bound. In fact, a similar result is proved for any
minor-closed property of bounded degree graphs.
As an immediate corollary of the above result we infer that many well studied
graph properties, like being planar, outer-planar, series-parallel, bounded
genus, bounded tree-width and several others, are testable with a constant
number of queries, where the constant may depend on and , but not
on the graph size. None of these properties was previously known to be testable
even with queries
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
Limits of Random Trees
Local convergence of bounded degree graphs was introduced by Benjamini and
Schramm. This result was extended further by Lyons to bounded average degree
graphs. In this paper, we study the convergence of a random tree sequence where
the probability of a given tree is proportional to . We show that this sequence is convergent and describe the limit
object, which is a random infinite rooted tree
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
What does the local structure of a planar graph tell us about its global structure?
The local k-neighborhood of a vertex v in an unweighted graph G = (V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G?
In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k = k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices)
Every property is testable on a natural class of scale-free multigraphs
In this paper, we introduce a natural class of multigraphs called
hierarchical-scale-free (HSF) multigraphs, and consider constant-time
testability on the class. We show that a very wide subclass, specifically, that
in which the power-law exponent is greater than two, of HSF is hyperfinite.
Based on this result, an algorithm for a deterministic partitioning oracle can
be constructed. We conclude by showing that every property is constant-time
testable on the above subclass of HSF. This algorithm utilizes findings by
Newman and Sohler of STOC'11. However, their algorithm is based on the
bounded-degree model, while it is known that actual scale-free networks usually
include hubs, which have a very large degree. HSF is based on scale-free
properties and includes such hubs. This is the first universal result of
constant-time testability on the general graph model, and it has the potential
to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and
SF become more general. Typos were fixe
How to solve the cake-cutting problem in sublinear time
In this paper, we show algorithms for solving the cake-cutting problem in
sublinear-time. More specifically, we preassign (simple) fair portions to o(n)
players in o(n)-time, and minimize the damage to the rest of the players. All
currently known algorithms require Omega(n)-time, even when assigning a portion
to just one player, and it is nontrivial to revise these algorithms to run in
-time since many of the remaining players, who have not been asked any
queries, may not be satisfied with the remaining cake. To challenge this
problem, we begin by providing a framework for solving the cake-cutting problem
in sublinear-time. Generally speaking, solving a problem in sublinear-time
requires the use of approximations. However, in our framework, we introduce the
concept of "eps n-victims," which means that eps n players (victims) may not
get fair portions, where 0< eps =< 1 is an arbitrary constant. In our
framework, an algorithm consists of the following two parts: In the first
(Preassigning) part, it distributes fair portions to r < n players in
o(n)-time. In the second (Completion) part, it distributes fair portions to the
remaining n-r players except for the eps n victims in poly}(n)-time. There are
two variations on the r players in the first part. Specifically, whether they
can or cannot be designated. We will then present algorithms in this framework.
In particular, an O(r/eps)-time algorithm for r =< eps n/127 undesignated
players with eps n-victims, and an O~(r^2/eps)-time algorithm for r =< eps
e^{{sqrt{ln{n}}}/{7}} designated players and eps =< 1/e with eps n-victims are
presented.Comment: 15 pages, no figur
A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Motivated by the problem of testing planarity and related properties, we
study the problem of designing efficient {\em partition oracles}. A {\em
partition oracle} is a procedure that, given access to the incidence lists
representation of a bounded-degree graph and a parameter \eps,
when queried on a vertex , returns the part (subset of vertices) which
belongs to in a partition of all graph vertices. The partition should be
such that all parts are small, each part is connected, and if the graph has
certain properties, the total number of edges between parts is at most \eps
|V|. In this work we give a partition oracle for graphs with excluded minors
whose query complexity is quasi-polynomial in 1/\eps, thus improving on the
result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition
oracle with query complexity exponential in 1/\eps. This improvement implies
corresponding improvements in the complexity of testing planarity and other
properties that are characterized by excluded minors as well as sublinear-time
approximation algorithms that work under the promise that the graph has an
excluded minor.Comment: 13 pages, 1 figur
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