For your first attempt at an itinerary, you might look at taking the cheapest flight out of San Francisco (let’s say it’s Seattle), then taking the cheapest flight from there to any of the other remaining cities (call it Los Angeles), then the cheapest from there (say, New York), and so forth, until you’re at your tenth city and you fly from there back to San Francisco. This is an example of a so-called greedy algorithm, which you can also think of as a “myopic algorithm”: one that shortsightedly takes the best thing available every step of the way.
