A Case for Strongly Polynomial Time Sub-Polyhedral Scheduling Using Two-Variable-Per-Inequality Polyhedra

We make a case for sub-polyhedral scheduling using (Unit-)Two-Variable-Per-Inequality or (U)TVPI Polyhedra. We empirically show that using general convex polyhedra leads to a scalability problem in a widely used polyhedral scheduler. We propose methods in which polyhedral schedulers can beat the scalability challenge by using sub-polyhedral under-aproximations of the polyhedra resulting from the application of the affine form of the Farkas lemma. We propose simple algorithms that under-approximate a general polyhedra into (U)TVPI polyhedra. These algorithms take worstcase polynomial time. We implement the above approximation algorithms in a modified PLuTo, and show that for a majority of the Polybench 2.0 kernels, the above under-approximation yield polyhedra that are non-empty. We also provide preliminary evidence that code generated by our sub-polyhedral parallelization prototype matches the performance of PLuTo-optimized code when the under-approximation preserves feasibility.