Two Geometric Optimization Problems

We consider two optimization problems with geometric structures The rst one con cerns the following minimization problem termed as the rectilinear polygon cover problem Cover certain features of a given rectilinear polygon possibly with rectilinear holes with the minimum number of rectangles included in the polygon Depending upon whether one wants to cover the interior boundary or corners of the polygon the problem is termed as the interior boundary or corner cover problem respectively Most of these problems are known to be NP complete In this chapter we survey some of the important previous results for these problems and provide a proof of impossibility of a polynomial time approximation scheme for the interior and boundary cover prob lems The second problem concerns routing in a segmented routing channel The related problems are fundamental to routing and design automation for Field Programmable Gate Arrays FPGAs a new type of electrically programmable VLSI In this chapter we survey the theoretical results on the combinatorial complexity and algorithm design for segmented channel routing It is known that the segmented channel routing problem is in general NP Complete E cient polynomial time algorithms for a number of important special cases are presented