Speed scaling for stretch plus energy

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