Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Size and the Approximability of Minimum Temporally Connected
  Subgraphs   We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with $n$
vertices and $\Omega(n^2)$ edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least $O(2^{\log^{1-\epsilon}
n})$ and at most $O(\min\{n^{1+\epsilon}, (\Delta M)^{2/3+\epsilon}\})$, for
any constant $\epsilon > 0$, where $M$ is the number of temporal edges and
$\Delta$ is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and $2$-approximable in cycles.

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The categories of this paper are: dg-ga
2
The categories of this paper are: math
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
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The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat