Mk Landscapes, NK Landscapes, MAX-kSAT: A Proof that the Only Challenging Problems are Deceptive

This paper investigates Gray Box Optimization for pseudo-Boolean optimization problems composed of M subfunctions, where each subfunction accepts at most k variables. We will refer to these as Mk Landscapes. In Gray Box optimization, the optimizer is given access to the set of M subfunctions. If the set of subfunctions is k-bounded and separable, the Gray Box optimizer is guaranteed to return the global optimum with 1 evaluation. A problem is said to be order k deceptive if the average values of hyperplanes over combinations of k variables cannot be used to infer a globally optimal solution. Hyperplane averages are always efficiently computable for Mk Landscapes. If a problem is not deceptive, the Gray Box optimizer also returns the global optimum after 1 evaluation. Finally, these concepts are used to understand the nonlinearity of problems in the complexity class P, such as Adjacent NK Landscapes. These ideas are also used to understand the problem structure of NP Hard problems such as MAX-kSAT and general Mk Landscapes. In general, NP Hard problems are profoundly deceptive.