The computational complexity of N-K fitness functions

N-K fitness landscapes have been used widely as examples and test functions in the field of evolutionary computation. We investigate the computational complexity of the problem of optimizing the N-K fitness functions and related fitness functions. We give an algorithm to optimize adjacent-model N-K fitness functions, which is polynomial in N. We show that the decision problem corresponding to optimizing random-model N-K fitness functions is NP-complete for K>1, and is polynomial for K=1. If the restriction that the ith component function depends on the ith bit is removed, then the problem is NP-complete, even for K=1. We also give a polynomial-time approximation algorithm for the arbitrary-model N-K optimization problem.