This paper studies the application of fixed-parameter tractable (FPT) algorithms to solve computer-aided design (CAD) problems. Specifically, we focus on layout decomposition problems for three lithography technologies: double patterning lithography (DPL), DPL with E-beam lithography (DPL+EBL), and DPL+DSA+EBL. Layout decomposition for the first two technologies are long-standing open problems without efficient optimal solutions, and the third technology is very promising in the future. The proposed approaches use ideas drastically different from all the previous works and can get optimal solutions in a short time. We show the great potential of applying FPT algorithms to solve more NP-hard problems efficiently in CAD.