On the Complexity of Single Row Routing Problems

We consider single row routing problems in VLSI. Previous work has left open, for all k≥3, whether or not there is an efficient algorithm to route interconnections on the minimum number of layers when only k channels are permitted per layer. We show that deciding whether such a routing exists is NP-complete even when k=3 and the interconnections are 2-pin nets. This solves an important open problem often cited in the literature. We also improve the running times of previously described algorithms for obtaining optimal single row routings by showing how to eliminate several paths in exhaustive search. Although this result still yields an exponential time algorithm (which presumably is necessary, as the problem is NP-complete), it solves routings instances with more channels than could be handled before.