Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Explicit linear kernels for packing problems   During the last years, several algorithmic meta-theorems have appeared
(Bodlaender et al. [FOCS 2009], Fomin et al. [SODA 2010], Kim et al. [ICALP
2013]) guaranteeing the existence of linear kernels on sparse graphs for
problems satisfying some generic conditions. The drawback of such general
results is that it is usually not clear how to derive from them constructive
kernels with reasonably low explicit constants. To fill this gap, we recently
presented [STACS 2014] a framework to obtain explicit linear kernels for some
families of problems whose solutions can be certified by a subset of vertices.
In this article we enhance our framework to deal with packing problems, that
is, problems whose solutions can be certified by collections of subgraphs of
the input graph satisfying certain properties. ${\mathcal F}$-Packing is a
typical example: for a family ${\mathcal F}$ of connected graphs that we assume
to contain at least one planar graph, the task is to decide whether a graph $G$
contains $k$ vertex-disjoint subgraphs such that each of them contains a graph
in ${\mathcal F}$ as a minor. We provide explicit linear kernels on sparse
graphs for the following two orthogonal generalizations of ${\mathcal
F}$-Packing: for an integer $\ell \geq 1$, one aims at finding either
minor-models that are pairwise at distance at least $\ell$ in $G$
($\ell$-${\mathcal F}$-Packing), or such that each vertex in $G$ belongs to at
most $\ell$ minors-models (${\mathcal F}$-Packing with $\ell$-Membership).
Finally, we also provide linear kernels for the versions of these problems
where one wants to pack subgraphs instead of minors.

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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc