Switchbox routing in VLSI design: Closing the complexity gap

The design of integrated circuits has achieved a great deal of attention in the last decade. In the routing phase, there have survived two unsolved layout problems that 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 also is possible to adopt our construction to channel routing implying that this problem is NP-complete, even if each net does not consist of more than five terminals. This improves upon a result of Sarrafzadeh who proved the NP-completeness in case of nets with no more than six terminals.