Approximating MIS over equilateral $B_1$-VPG graphs.

We present an approximation algorithm for the maximum independent set (MIS) problem over the class of equilateral $B_1$-VPG graphs. These are intersection graphs of $L$-shaped planar objects % (and their rotations by multiples of $90^o$) with both arms of each object being equal. We obtain a $36(\log 2d)$-approximate algorithm running in $O(n(\log n)^2)$ time for this problem, where $d$ is the ratio $d_{max}/d_{min}$ and $d_{max}$ and $d_{min}$ denote respectively the maximum and minimum length of any arm in the input equilateral $L$-representation of the graph. In particular, we obtain $O(1)$-factor approximation of MIS for $B_1$-VPG -graphs for which the ratio $d$ is bounded by a constant. % formed by unit length $L$-shapes. In fact, algorithm can be generalized to an $O(n(\log n)^2)$ time and a $36(\log 2d_x)(\log 2d_y)$-approximate MIS algorithm over arbitrary $B_1$-VPG graphs. Here, $d_x$ and $d_y$ denote respectively the analogues of $d$ when restricted to only horizontal and vertical arms of members of the input. This is an improvement over the previously best $n^\epsilon$-approximate algorithm \cite{FoxP} (for some fixed $\epsilon>0$), unless the ratio $d$ is exponentially large in $n$. In particular, $O(1)$-approximation of MIS is achieved for graphs with $\max\{d_x,d_y\}=O(1)$.