105,869 research outputs found
Orbit counting in conjugacy classes for free groups acting on trees
In this paper we study the action of the fundamental group of a finite metric
graph on its universal covering tree. We assume the graph is finite, connected
and the degree of each vertex is at least three. Further, we assume an
irrationality condition on the edge lengths. We obtain an asymptotic for the
number of elements in a fixed conjugacy class for which the associated
displacement of a given base vertex in the universal covering tree is at most
. Under a mild extra assumption we also obtain a polynomial error term.Comment: 13 pages, additional section discusses error terms, revised
expositio
Large deviations of empirical neighborhood distribution in sparse random graphs
Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is
present independently with probability c/n, with c>0 fixed. For large n, a
typical random graph locally behaves like a Galton-Watson tree with Poisson
offspring distribution with mean c. Here, we study large deviations from this
typical behavior within the framework of the local weak convergence of finite
graph sequences. The associated rate function is expressed in terms of an
entropy functional on unimodular measures and takes finite values only at
measures supported on trees. We also establish large deviations for other
commonly studied random graph ensembles such as the uniform random graph with
given number of edges growing linearly with the number of vertices, or the
uniform random graph with given degree sequence. To prove our results, we
introduce a new configuration model which allows one to sample uniform random
graphs with a given neighborhood distribution, provided the latter is supported
on trees. We also introduce a new class of unimodular random trees, which
generalizes the usual Galton Watson tree with given degree distribution to the
case of neighborhoods of arbitrary finite depth. These generalized Galton
Watson trees turn out to be useful in the analysis of unimodular random trees
and may be considered to be of interest in their own right.Comment: 58 pages, 5 figure
Parameterized Complexity of Secluded Connectivity Problems
The Secluded Path problem models a situation where a sensitive information
has to be transmitted between a pair of nodes along a path in a network. The
measure of the quality of a selected path is its exposure, which is the total
weight of vertices in its closed neighborhood. In order to minimize the risk of
intercepting the information, we are interested in selecting a secluded path,
i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem
is to find a tree in a graph connecting a given set of terminals such that the
exposure of the tree is minimized. The problems were introduced by Chechik et
al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded
Path is fixed-parameter tractable (FPT) on unweighted graphs being
parameterized by the maximum vertex degree of the graph and that Secluded
Steiner Tree is FPT parameterized by the treewidth of the graph. In this work,
we obtain the following results about parameterized complexity of secluded
connectivity problems.
We give FPT-algorithms deciding if a graph G with a given cost function
contains a secluded path and a secluded Steiner tree of exposure at most k with
the cost at most C.
We initiate the study of "above guarantee" parameterizations for secluded
problems, where the lower bound is given by the size of a Steiner tree.
We investigate Secluded Steiner Tree from kernelization perspective and
provide several lower and upper bounds when parameters are the treewidth, the
size of a vertex cover, maximum vertex degree and the solution size. Finally,
we refine the algorithmic result of Chechik et al. by improving the exponential
dependence from the treewidth of the input graph.Comment: Minor corrections are don
Finding Cycles and Trees in Sublinear Time
We present sublinear-time (randomized) algorithms for finding simple cycles
of length at least and tree-minors in bounded-degree graphs. The
complexity of these algorithms is related to the distance of the graph from
being -minor-free (resp., free from having the corresponding tree-minor).
In particular, if the graph is far (i.e., -far) {from} being
cycle-free, i.e. if one has to delete a constant fraction of edges to make it
cycle-free, then the algorithm finds a cycle of polylogarithmic length in time
\tildeO(\sqrt{N}), where denotes the number of vertices. This time
complexity is optimal up to polylogarithmic factors.
The foregoing results are the outcome of our study of the complexity of {\em
one-sided error} property testing algorithms in the bounded-degree graphs
model. For example, we show that cycle-freeness of -vertex graphs can be
tested with one-sided error within time complexity
\tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known
query lower bound, and contrasts with the fact that any minor-free property
admits a {\em two-sided error} tester of query complexity that only depends on
the proximity parameter \e. For any constant , we extend this result
to testing whether the input graph has a simple cycle of length at least .
On the other hand, for any fixed tree , we show that -minor-freeness has
a one-sided error tester of query complexity that only depends on the proximity
parameter \e.
Our algorithm for finding cycles in bounded-degree graphs extends to general
graphs, where distances are measured with respect to the actual number of
edges. Such an extension is not possible with respect to finding tree-minors in
complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree
Graphs, One-Sided vs Two-Sided Error Probability Updated versio
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by NeÅ”etÅil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of NeÅ”etÅil and Ossona de Mendez on the existence of low tree-depth colorings
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