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On fundamental tradeoffs between delay bounds and computational complexity in packet scheduling algorithms

We clarify, extend, and solve a long-standing 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(log n). We also discover that, surprisingly, the complexity lower bound remains the same even if the delay bound is relaxed to O(na) for O < a < 1. This implies that the delay-complexity tradeoff curve is flat in the "interval" [O(1), O(n)). We later conditionally extend both complexity results (for O(1) or O(na) delay) to a much stronger computational model, the linear decision tree. Finally, we show that the same complexity lower bounds are conditionally applicable to guaranteeing tight end-to-end delay bounds, if the delay bounds are provided through the Latency Rate (LR) framework.