  A graph $G$ is called $B_k$-VPG, for some constant $k\geq 0$, if it has a
string representation on an axis-parallel grid such that each vertex is a path
with at most $k$ bends and two vertices are adjacent in $G$ if and only if the
corresponding paths intersect each other. The part of a path that is between
two consecutive bends is called a segment of the path. In this paper, we study
the Maximum-Weighted Independent Set problem on $B_k$-VPG graphs. The problem
is known to be NP-complete on $B_1$-VPG graphs, even when the two segments of
every path have unit length [12], and $O(\log n)$-approximation algorithms are
known on $B_k$-VPG graphs, for $k\leq 2$ [3, 14]. In this paper, we give a
$(ck+c+1)$-approximation algorithm for the problem on $B_k$-VPG graphs for any
$k\geq 0$, where $c>0$ is the length of the longest segment among all segments
of paths in the graph. Notice that $c$ is not required to be a constant; for
instance, when $c\in O(\log \log n)$, we get an $O(\log \log n)$-approximation
or we get an $O(1)$-approximation when $c$ is a constant. To our knowledge,
this is the first $o(\log n)$-approximation algorithm for a non-trivial
subclass of $B_k$-VPG graphs.
