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In this paper, we introduce and study the Rectangle Escape Problem (REP), which is motivated by PCB bus escape routing. Given a rectangular region R and a set S of rectangles within R, the REP is to choose a direction for each rectangle to escape to the boundary of R, such that the resultant maximum density over R is minimized. We prove that the REP is NP-Complete, and show that it can be formulated as an Integer Linear Program (ILP). A provably good approximation algorithm for the REP is developed by applying Linear Programming (LP) relaxation and a special rounding technique to the ILP. This approximation algorithm is also shown to work for a more general version of REP with weights (weighted REP). In addition, an iterative refinement procedure is proposed as a postprocessing step to further improve the results. Our approach is tested on a set of industrial PCB bus escape routing problems. Experimental results show that the optimal solution can be obtained within 3 seconds for each of the test cases.