Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Hardness of Approximating the Threshold Dimension, Boxicity and
  Cubicity of a Graph   A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times ...
\times R_k$ where each $R_i$ is a closed interval on the real line. The {\it
boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$
such that $G$ can be represented as the intersection graph of a collection of
$k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is
defined as the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where
each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$.
The {\it cubicity} of $G$, denoted as $\cub(G)$, is the minimum integer $k$
such that $G$ can be represented as the intersection graph of a collection of
$k$-cubes. The {\it threshold dimension} of a graph $G(V,E)$ is the smallest
integer $k$ such that $E$ can be covered by $k$ threshold spanning subgraphs of
$G$. In this paper we will show that there exists no polynomial-time algorithm
to approximate the threshold dimension of a graph on $n$ vertices with a factor
of $O(n^{0.5-\epsilon})$ for any $\epsilon >0$, unless $NP=ZPP$. From this
result we will show that there exists no polynomial-time algorithm to
approximate the boxicity and the cubicity of a graph on $n$ vertices with
factor $O(n^{0.5-\epsilon})$ for any $ \epsilon >0$, unless $NP=ZPP$. In fact
all these hardness results hold even for a highly structured class of graphs
namely the split graphs. We will also show that it is NP-complete to determine
if a given split graph has boxicity at most 3.

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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph