We are pilots, who approach the OP question with technical answers. The real answer is that airlines don't actually worry about saving fuel. They want to save money!. Fuel and money may not be the same thing;sometimes. At my company, right now, they are attempting to save money.
We fly the flight plan with single engine taxi when we can. I get ACARS messages sometimes asking me to slow down, on a transcon, to allow better flow at the Hub for a Hub bank. We fly high speed sometimes because they want the airplane somewhere so another crew can get another hour of utilization that day.
There is a giant computer system in Dallas that makes these decisions along with folks in the System Control Center. I just fly the airplane they way they tell me, mostly, and let them worry about the big picture of the airline. There are almost a thousand airplanes and an average of nearly 6,700 flights per day to nearly 350 destinations in more than 50 countries.
It's sort of like the Army...Shut up and do like you are told. The only difference is it pays better and I don't have to salute anyone and we don't have bombs or heavy artillery.