On fundamental tradeoffs between delay bounds and computational complexity in packet scheduling algorithms
Jun Jim Xu, Richard J. Lipton · 2002
In this work, we clarify, extend and solve an open problem concerning the computational complexity for packet scheduling algorithms to achieve tight end-to-end delay bounds. We first focus on the difference between the time a packet finishes service in a scheduling algorithm and its virtual finish time under a GPS (General Processor Sharing) scheduler, called GPS-relative delay. We prove that, under a slightly restrictive but reasonable computational model, the lower bound computational complexity of any scheduling algorithm that guarantees O(1) GPS-relative delay bound is Ω (log2 n) (widely believed as a "folklore theorem" but never proved). We also discover that, surprisingly, the complexity lower bound remains the same even if the delay bound is relaxed to O(na) for 0‹a⋵1. This implies that the delay-complexity tradeoff curve is "flat" in the "interval" [O(1), O(n)). We later extend both complexity results (for O(1) or O(na) delay) to a much stronger computational model. Finally, we show that the same complexity lower bounds are conditionally applicable to guaranteeing tight end-to-end delay bounds. This is done by untangling the relationship between the GPS-relative delay bound and the end-to-end delay bound.