The shortest distance between two points is a straight line i.e. a great circle track. The shortest distance will normally result in flying for the least time to cover that distance which will result in the minimum time in the air. As I stated, other factors come into play which may make it more economical to fly a less direct route. Airlines have departments devoted to flight planning and take all these factors into account to come up with the most economical route.
The minimum time in the air may not result in the lowest cost if heavy overflight charges are involved, it may be cheaper to spend a little longer in the air and fly around. North Korea charges very expensive fees to overfly the country, airlines decide whether to pay or take an alternative route.
The cost of operating the aircraft also comes into play, if fuel is cheap as it is at the moment then extra flying time costs less. The hourly lease rate varies among aircraft as well, it may be cheaper to speed up and use more fuel or slow down and use less fuel. If fuel is expensive and there is a tailwind it may be more economical to reduce cruise speed and ride the wind. Crew flight time limits are also a consideration, if a return flight with the same crew is required then it may be cheaper to fly fast to stay within limits, if a night stop is involved timing is less critical.
Minimum time may not necessarily result in the minimum cost.