Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Kernelization and Approximation for the Vector Connectivity Problem   In the Vector Connectivity problem we are given an undirected graph
$G=(V,E)$, a demand function $\phi\colon V\to\{0,\ldots,d\}$, and an integer
$k$. The question is whether there exists a set $S$ of at most $k$ vertices
such that every vertex $v\in V\setminus S$ has at least $\phi(v)$
vertex-disjoint paths to $S$; this abstractly captures questions about placing
servers or warehouses relative to demands. The problem is \NP-hard already for
instances with $d=4$ (Cicalese et al., arXiv '14), admits a log-factor
approximation (Boros et al., Networks '14), and is fixed-parameter tractable in
terms of~$k$ (Lokshtanov, unpublished '14). We prove several results regarding
kernelization and approximation for Vector Connectivity and the variant Vector
$d$-Connectivity where the upper bound $d$ on demands is a fixed constant. For
Vector $d$-Connectivity we give a factor $d$-approximation algorithm and
construct a vertex-linear kernelization, i.e., an efficient reduction to an
equivalent instance with $f(d)k=O(k)$ vertices. For Vector Connectivity we have
a factor $\text{opt}$-approximation and we can show that it has no
kernelization to size polynomial in $k$ or even $k+d$ unless
$\mathsf{NP\subseteq coNP/poly}$, making $f(d)\operatorname{poly}(k)$ optimal
for Vector $d$-Connectivity. Finally, we provide a write-up for fixed-parameter
tractability of Vector Connectivity($k$) by giving an alternative FPT algorithm
based on matroid intersection.

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The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: math
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The categories of this paper are: bayes-an
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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: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int