Kernelization and Sparseness: the case of Dominating Set

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-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 $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-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

Sting jets in intense winter North-Atlantic windstorms

Extratropical cyclones dominate autumn and winter weather over western Europe. The strongest cyclones, often termed windstorms, have a large socio-economic impact due to the strong surface winds and associated storm surges in coastal areas. Here we show that sting jets are a common feature of windstorms; up to a third of the 100 most intense North Atlantic winter windstorms over the last two decades satisfy conditions for sting jets. The sting jet is a mesoscale descending airstream that can cause strong near-surface winds in the dry slot of the cyclone, a region not usually associated with strong winds. Despite their localized transient nature these sting jets can cause significant damage, a prominent example being the storm that devastated southeast England on 16 October 1987. We present the first regional climatology of windstorms with sting jets. Previously analysed sting jet cases appear to have been exceptional in their track over northwest Europe rather than in their strength

A comparison of two identification and tracking methods for polar lows

In this study, we compare two different cyclone-tracking algorithms to detect North Atlantic polar lows, which are very intense mesoscale cyclones. Both approaches include spatial filtering, detection, tracking and constraints specific to polar lows. The first method uses digital bandpass-filtered mean sea level pressure (MSLP) fieldsin the spatial range of 200�600 km and is especially designed for polar lows. The second method also uses a bandpass filter but is based on the discrete cosine transforms (DCT) and can be applied to MSLP and vorticity fields. The latter was originally designed for cyclones in general and has been adapted to polar lows for this study. Both algorithms are applied to the same regional climate model output fields from October 1993 to September 1995 produced from dynamical downscaling of the NCEP/NCAR reanalysis data. Comparisons between these two methods show that different filters lead to different numbers and locations of tracks. The DCT is more precise in scale separation than the digital filter and the results of this study suggest that it is more suited for the bandpass filtering of MSLP fields. The detection and tracking parts also influence the numbers of tracks although less critically. After a selection process that applies criteria to identify tracks of potential polar lows, differences between both methods are still visible though the major systems are identified in both

A c k n 5-Approximation Algorithm for Treewidth

We give an algorithm that for an input n-vertex graph G and integer k> 0, in time O(c k n) either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single-exponential in k and linear in n. Treewidth based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single-exponential in the treewidth and linear in the input size

A cκn 5-Approximation algorithm for treewidth

We give an algorithm that for an input n-vertex graph G and integer κ > 0, in time 2O(κ)n, either outputs that the treewidth of G is larger than κ, or gives a tree decomposition of G of width at most 5κ + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in κ and linear in n. Treewidth-based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single exponential in the treewidth and linear in the input size

A c\u3csup\u3eκ\u3c/sup\u3en 5-Approximation algorithm for treewidth

\u3cp\u3eWe give an algorithm that for an input n-vertex graph G and integer κ > 0, in time 2\u3csup\u3eO(κ)\u3c/sup\u3en, either outputs that the treewidth of G is larger than κ, or gives a tree decomposition of G of width at most 5κ + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in κ and linear in n. Treewidth-based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single exponential in the treewidth and linear in the input size.\u3c/p\u3

A cκn 5-Approximation algorithm for treewidth

We give an algorithm that for an input n-vertex graph G and integer κ > 0, in time 2O(κ)n, either outputs that the treewidth of G is larger than κ, or gives a tree decomposition of G of width at most 5κ + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in κ and linear in n. Treewidth-based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single exponential in the treewidth and linear in the input size