On the k-layer planar subset and via minimization problems

An important problem in performance-driven routing is the k -layer planar subset problem which is to choose a maximum (weighted) subset of nets such that each net in the subset can be routed in one of k `preferred´ layers. Related to the k-layer planar subset problem is the k-layer topological via-minimization problem which is to determine the topology of each net using k routing layers such that a minimum number of vias is used. For the case k=2, the topological via minimization problem has been studied by CAD researchers for a long time because of its practical and theoretical importance. The authors show that both the general k-layer planar subset problem and the k-layer topological via minimisation problem are NP-complete. Moreover, they show that both problems can be solved in polynomial time for a large class of channels, called crossing channels. It can be shown that under a realistic assumption, all the channels for inter-block connections in the general cell design style are crossing channels. The authors algorithms are based on an efficient algorithm for computing a maximum weighted·k-cofamily in a partially ordered set