Speed scaling for stretch plus energy

We consider speed scaling problems where the objective is to minimize a linear combination of arbitrary scheduling objective SS, and energy EE. A natural conjecture is that there is an O(1)O(1)-competitive algorithm for SS on a fixed speed processor if and only if there is an O(1)O(1)-competitive algorithm for S+ES+E on a processor with an arbitrary power function. We give evidence to support this conjecture by providing an O(1)O(1)-competitive algorithm for the objective of integer stretch plus energy.