Topological via minimization revisited

The topological via minimization problem in a two-layer environment is considered. A set of n two-terminal nets in a bounded region is given. The authors attempt to find a homotopy to assign nets to distinct layers so that no two nets on the same layer cross each other and the number of vias is minimized. A recursive approach in which an optimal solution to a two-sided channel routing problem is used as a basis is used to solve this problem optimally. The notion of partition number K of a circle graph is introduced, and the total running time of the via minimization algorithm is shown to be O((n/K)^2K-2 log (n/K)), where n is the total number of nets