A Constant Factor Approximation Algorithm for Boxicity of Circular Arc   Graphs
  Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - \epsilon})$-factor, for any $\epsilon >0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^\epsilon$-factor approximation algorithm for computing boxicity is known, for any $\epsilon <1$. In this paper, we study the boxicity problem on Circular Arc graphs - intersection graphs of arcs of a circle. We give a $(2+\frac{1}{k})$-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where $k \ge 1$ is its boxicity. For Normal Circular Arc(NCA) graphs, with an NCA model given, this can be improved to an additive 2-factor approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity is $O(mn+n^2)$ in both these cases and in $O(mn+kn^2)= O(n^3)$ time we also get their corresponding box representations, where $n$ is the number of vertices of the graph and $m$ is its number of edges. The additive 2-factor algorithm directly works for any Proper Circular Arc graph, since computing an NCA model for it can be done in polynomial time. 
Abhijin Adiga and Jasine Babu and L. Sunil Chandran
cs.DS cs.DM math.CO