Orthogonal dissection into few rectangles

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.Comment: 18 pages, 8 figures. This version adds results on dissection with rotations and reflection

Applications of mathematical network theory

This thesis is a collection of papers on a variety of optimization problems where network structure can be used to obtain efficient algorithms. The considered applications range from the optimization of radiation treatment plkans in cancer therapy to maintenance planning for maximizing the throughput in bulk good supply chains