Rethinking Tractability for Schedulability Analysis

Algorithms that have been developed for solving computationally intractable schedulability analysis problems may be classified into two broad categories: exact algorithms that run in exponential time, and polynomial-time algorithms that provide approximate solutions. If exact algorithms are sought, it has traditionally been required that these algorithms have pseudo-polynomial running time. More recently, schedulability analysis algorithms that have polynomial running time but are allowed to make calls to an ILP solver have increasingly been considered tractable. When approximation algorithms are acceptable, an objective has been to obtain Fully Polynomial-Time Approximation Schemes, which are 'tunable' algorithms that provide a smooth transition between polynomial time and exponential time by letting the user of the algorithm set an appropriate value for a parameter. In this paper we take a fresh view on the connections between the various perspectives on what is considered to be tractable schedulability analysis. We seek to determine when the different forms of tractable analyses are applicable to a particular problem and what problem features rules them out, and demonstrate our findings upon concrete scheduling problems. We also suggest that 'pseudo-polynomial time' is perhaps a rather broad category, and propose a finer-grained classification of the class of pseudo-polynomial time algorithms.