10 research outputs found
Testing bounded arboricity
In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are -close to having
arboricity and graphs that -far from having
arboricity , where is an absolute small constant. The query
complexity and running time of the algorithm are
where denotes
the number of vertices and denotes the number of edges. In terms of the
dependence on and this bound is optimal up to poly-logarithmic factors
since queries are necessary (and .
We leave it as an open question whether the dependence on can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Tight approximation bounds for dominating set on graphs of bounded arboricity
We consider the minimum dominating set problem on graphs with bounded arboricity. For graphs with arboricity a, we give an LP rounding algorithm that achieves an approximation factor of 3a and we complement this with a matching (up to constants) hardness of approximation result
Tight approximation bounds for dominating set on graphs of bounded arboricity
We consider the minimum dominating set problem on graphs with bounded arboricity. For graphs with arboricity a, we give an LP rounding algorithm that achieves an approximation factor of 3a and we complement this with a matching (up to constants) hardness of approximation result