Switchbox routing in VLSI design: Closing the complexity gap

Abstract

The design of integrated circuits has achieved a great deal of attention in the last decade. In the routing phase, there have survived two open layout problems which are important from both the theoretical and the practical point of view. Up to now, switchbox routing has been known to be solvable in polynomial time when there are only 2-terminal nets, and to be NP-complete in case there exist nets involving at least five terminals. Our main result is that this problem is NP-complete even if no net has more than three terminals. Hence, from the theoretical perspective, the switchbox routing problem is completely settled. The NP-completeness proof is based on a reduction from a special kind of the satisfiability problem. It is also possible to adopt our construction to channel routing which shows that this problem is NP-complete, even if each net does not consist of more than four terminals. This improves upon a result of Sarrafzadeh who showed the NP-completeness in case of nets with no more than five terminals.