Some polynomially solvable subcases of the detailed routing problem in VLSI design

There are plenty of NP-complete problems in very large scale integrated design, like channel routing or switchbox routing in the two-layer Manhattan model (2Mm, for short). However, there are quite a few polynomially solvable problems as well. Some of them (like the single row routing in 2Mm) are "classical" results; in a past survey [36] we presented some more recent ones, including: 1. a linear time channel routing algorithm in the unconstrained two-layer model; 2. a linear time switchbox routing algorithm in the unconstrained multilayer model; and 3. a linear time solution of the so called gamma routing problem in 2Mm. (This latter means that all the terminals to be interconnected are situated at two adjacent sides of a rectangular routing area, thus forming a Gamma shape. Just like channel routing, it is a special case of switchbox routing, and contains single row routing as a special case.) In the present survey talk we also mention some results from the last three years, including 4. some negative results (NP-completeness) in the multilayer Manhattan model and a channel routing algorithm if the number of layers is even; 5. an interesting relation between channel routing and multiprocessor scheduling; and 6. some improvements of 1-3 above. We present the (positive and negative) results in a systematic way, taking into account two hierarchies, namely that of geometry (what to route') and technology (how to route) at the same time. (C) 2001 Elsevier Science B.V. All rights reserved.