18,186 research outputs found
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
Graph cluster randomization: network exposure to multiple universes
A/B testing is a standard approach for evaluating the effect of online
experiments; the goal is to estimate the `average treatment effect' of a new
feature or condition by exposing a sample of the overall population to it. A
drawback with A/B testing is that it is poorly suited for experiments involving
social interference, when the treatment of individuals spills over to
neighboring individuals along an underlying social network. In this work, we
propose a novel methodology using graph clustering to analyze average treatment
effects under social interference. To begin, we characterize graph-theoretic
conditions under which individuals can be considered to be `network exposed' to
an experiment. We then show how graph cluster randomization admits an efficient
exact algorithm to compute the probabilities for each vertex being network
exposed under several of these exposure conditions. Using these probabilities
as inverse weights, a Horvitz-Thompson estimator can then provide an effect
estimate that is unbiased, provided that the exposure model has been properly
specified.
Given an estimator that is unbiased, we focus on minimizing the variance.
First, we develop simple sufficient conditions for the variance of the
estimator to be asymptotically small in n, the size of the graph. However, for
general randomization schemes, this variance can be lower bounded by an
exponential function of the degrees of a graph. In contrast, we show that if a
graph satisfies a restricted-growth condition on the growth rate of
neighborhoods, then there exists a natural clustering algorithm, based on
vertex neighborhoods, for which the variance of the estimator can be upper
bounded by a linear function of the degrees. Thus we show that proper cluster
randomization can lead to exponentially lower estimator variance when
experimentally measuring average treatment effects under interference.Comment: 9 pages, 2 figure
Recovering sparse graphs
We construct a fixed parameter algorithm parameterized by d and k that takes
as an input a graph G' obtained from a d-degenerate graph G by complementing on
at most k arbitrary subsets of the vertex set of G and outputs a graph H such
that G and H agree on all but f(d,k) vertices.
Our work is motivated by the first order model checking in graph classes that
are first order interpretable in classes of sparse graphs. We derive as a
corollary that if G_0 is a graph class with bounded expansion, then the first
order model checking is fixed parameter tractable in the class of all graphs
that can obtained from a graph G from G_0 by complementing on at most k
arbitrary subsets of the vertex set of G; this implies an earlier result that
the first order model checking is fixed parameter tractable in graph classes
interpretable in classes of graphs with bounded maximum degree
A PTAS for the minimum dominating set problem in unit disk graphs
We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. \u
Approximating Minimum Independent Dominating Sets in Wireless Networks
We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in a graph that is also independent. It thus combines the advantages of both structures, and there are many applications that rely on these two structures e.g. in the area of wireless ad hoc networks. The presented approach yields a robust algorithm, that is, the algorithm accepts any undirected graph as input, and returns a (1+")- pproximate minimum dominating set, or a certificate showing that the input graph does not reflect a wireless network
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