Two Geometric Optimization Problems

Abstract

We consider two optimization problems with geometric structures. The first one concerns the foUowing 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 find provide a proof of impossibility of a polynomial-time approximation scheme for the interior and boundary cover problems. 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. Efficient polynomial time algorithms for a number of important special cases are presented.