Register Allocation : What does the NP - completeness Proof of Chaitin et al . Really Prove ?

Register allocation is one of the most studied problems in compilation. It is considered as an NP-complete problem since Chaitin et al., in 1981, modeled the problem of assigning temporary variables to k machine registers as the problem of coloring, with k colors, the interference graph associated to the variables. The fact that the interference graph can be arbitrary proves the NP-completeness of this formulation. However, this original proof does not really show where the complexity of register allocation comes from. Recently, the re-discovery that interference graphs of SSA programs can be colored in polynomial time raised the question: Can we exploit SSA form to perform register allocation in polynomial time, without contradicting Chaitin et al’s NP-completeness result? To address such a question and, more generally, the complexity of register allocation, we revisit Chaitin et al’s proof to better identify the interactions between spilling (load/store insertion), coalescing/splitting (removal/insertion of moves between registers), critical edges (a property of the control-flow graph), and coloring (assignment to registers). In particular, we show that, in general (we will make clear when), it is easy to decide if temporary variables can be assigned to k registers or if some spilling is necessary. In other words, the real complexity does not come from the coloring itself (as a wrong interpretation of the proof of Chaitin et al. may suggest) but comes from the presence of critical edges and from the optimizations of spilling and coalescing.