  Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer
$k$ such that $G$ can be represented as the intersection graph of axis parallel
boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable
even for graph classes like bipartite, co-bipartite and split graphs within
$O(n^{1 - \epsilon})$-factor, for any $\epsilon >0$ in polynomial time unless
$NP=ZPP$. We give FPT approximation algorithms for computing the boxicity of
graphs, where the parameter used is the vertex or edge edit distance of the
given graph from families of graphs of bounded boxicity. This can be seen as a
generalization of the parameterizations discussed in \cite{Adiga2}.
  Extending the same idea in one of our algorithms, we also get an
$O\left(\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right)$ factor approximation
algorithm for computing boxicity and an $O\left(\frac{n {(\log \log
n)}^{\frac{3}{2}}}{\sqrt{\log n}}\right)$ factor approximation algorithm for
computing the cubicity. These seem to be the first $o(n)$ factor approximation
algorithms known for both boxicity and cubicity. As a consequence of this
result, a $o(n)$ factor approximation algorithm for computing the partial order
dimension of finite posets and a $o(n)$ factor approximation algorithm for
computing the threshold dimension of split graphs would follow.
