  In this paper we consider the Maximum Independent Set problem (MIS) on
$B_1$-EPG graphs. EPG (for Edge intersection graphs of Paths on a Grid) was
introduced in ~\cite{edgeintersinglebend} as the class of graphs whose vertices
can be represented as simple paths on a rectangular grid so that two vertices
are adjacent if and only if the corresponding paths share at least one edge of
the underlying grid. The restricted class $B_k$-EPG denotes EPG-graphs where
every path has at most $k$ bends. The study of MIS on $B_1$-EPG graphs has been
initiated in~\cite{wadsMIS} where authors prove that MIS is NP-complete on
$B_1$-EPG graphs, and provide a polynomial $4$-approximation. In this article
we study the approximability and the fixed parameter tractability of MIS on
$B_1$-EPG. We show that there is no PTAS for MIS on $B_1$-EPG unless P$=$NP,
even if there is only one shape of path, and even if each path has its vertical
part or its horizontal part of length at most $3$. This is optimal, as we show
that if all paths have their horizontal part bounded by a constant, then MIS
admits a PTAS. Finally, we show that MIS is FPT in the standard
parameterization on $B_1$-EPG restricted to only three shapes of path, and
$W_1$-hard on $B_2$-EPG. The status for general $B_1$-EPG (with the four
shapes) is left open.
