Asymptotic Behavior of Linear Approximations of Pseudo-Boolean Functions

We study the problem of approximating pseudo-Boolean functions by linear pseudo-Boolean functions. Pseudo-Boolean functions generalize ordinary Boolean functions by allowing the function values to be real numbers instead of just the 0-1 values. Pseudo-Boolean functions have been used by AI and theorem proving researchers for efficient constraint satisfaction solving. They can also be applied for modeling uncertainty. We investigate the possibility of efficiently computing a linear approximation of a pseudo-Boolean function of arbitrary degree. We show some example cases in which a simple (efficiently computable) linear approximating function works just as well as the best linear approximating function, which may require an exponential amount of computation to obtain. We conjecture that for any pseudo-Boolean function of fixed degree <I>k</I> >1 where <I>k</I> is independent of the number of Boolean variables, the best linear approximating function works better than simply using the linear part of the target function. We also study the behavior of the expected best linear approximating function when the target pseudo-Boolean function to be approximated is random.