#p-np

The Navier-Stokes equations—despite being exceptionally useful for modeling the weather, ocean currents, pipes, cars, airplane wings and other hydrodynamic systems, and despite the million-dollar prize offered for their exact solutions—are incomplete. Truer equations of fluid dynamics can be found in a little-known, relatively unheralded theory developed by the Dutch mathematician and physicist Diederik Korteweg in the early 1900s. And yet, for some gases, even the Korteweg equations fall short, and there is no fluid picture at all.

Randomization can help in attempts to solve the traveling salesman problem, asking for the shortest possible route visiting a number of given cities.

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The P-NP problem is the most important open problem in computer science, if not all of mathematics. Simply stated, it asks whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer. The Golden Ticket provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. In this informative and entertaining book, Lance Fortnow traces how the problem arose during the Cold War on both sides of the Iron Curtain, and gives examples of the problem from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. But difficulty also has its advantages. Hard problems allow us to safely conduct electronic commerce and maintain privacy in our online lives.

The Millennium Problem P versus NP is related to this concept, as it asks whether or not one can find a solution to a problem in polynomial time (deterministically) if one can check a solution in polynomial time.