The Hardness of Approximating the Threshold Dimension, Boxicity and Cubicity of a Graph

A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times … \times R_k$whereeach$R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\text{boxi}(G)$, 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 $R_1\times R_2\times \mathrm{.}\mathrm{.}\mathrm{.}\times R_k$ 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 $\text{cub}(G)$, 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 $O(n^{0.5-ϵ})$ for any $ϵ>0$, 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 $O(n^{0.5-ϵ})$ for any $ϵ>0$, 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.