On the k-layer planar subset and topological 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. In this paper, we show that both the general k -layer planar subset problem and the k -layer topological via minimization problem are NP-complete. Moreover, we show that both problems can be solved in polynomial time when the routing regions are crossing channels. It can be shown that under a suitable assumption, all the channels for inter-block connections in the general cell design style are crossing channels. Our algorithms are based on an efficient algorithm for computing a maximum weighted k -cofamily in a partially ordered set.