  Motivated by the applications of routing in PCB buses, the Rectangle Escape
Problem was recently introduced and studied. In this problem, we are given a
set of rectangles $\mathcal{S}$ in a rectangular region $R$, and we would like
to extend these rectangles to one of the four sides of $R$. Define the density
of a point $p$ in $R$ as the number of extended rectangles that contain $p$.
The question is then to find an extension with the smallest maximum density.
  We consider the problem of maximizing the number of rectangles that can be
extended when the maximum density allowed is at most $d$. It is known that this
problem is polynomially solvable for $d = 1$, and NP-hard for any $d \geq 2$.
We consider approximation and exact algorithms for fixed values of $d$. We also
show that a very special case of this problem, when all the rectangles are unit
squares from a grid, continues to be NP-hard for $d = 2$.
