On the complexity of loop fusion

Loop fusion is a program transformation that combines several loops into one. It is used in parallelizing compilers mainly for increasing the granularity of loops and for improving data reuse. The goal of this paper is to study, from a theoretical point of view, several variants of the loop fusion problem-identifying polynomially solvable cases and NP-complete cases-and to make the link between these problems and some scheduling problems that arise from completely different areas. We study among others, the fusion of loops of different types, and the fusion of loops when combined with loop shifting.