Quadratization of symmetric pseudo-Boolean functions

We consider the problem of minimizing an arbitrary pseudo-Boolean function f(x), that is, a real-valued function of 0-1 variables. In recent years, several authors have proposed to reduce this problem to the quadratic case by expressing f(x) as min{g(x,y):y∈{0,1}^m}, where g(x,y) is a quadratic pseudo-Boolean function of x and of additional binary variables y. We say that g(x,y) is a quadratization of f. In this talk, we investigate the number of additional variables needed in a quadratization when f is a symmetric function of the x-variables. The cases where f is either a positive or a negative monomial are of particular interest, but some of our techniques also extend to more complex functions, like k-out-of-n or parity functions.
Joint work with Martin Anthony, Endre Boros and Aritanan Gruber