Quadratization of symmetric pseudo-Boolean functions

A pseudo-Boolean function is a real-valued function f(x) = f(x1; x2; : : : ; xn) of n binary variables, that is, a mapping from f0; 1gn to R. For a pseudo-Boolean function f(x) on f0; 1gn, we say that g(x; y) is a quadratization of f if g(x; y) is a quadratic polynomial depending on x and on m auxiliary binaryvariables y1; y2; : : : ; ym such that f(x) = minfg(x; y) : y 2 f0; 1gmg forall x 2 f0; 1gn. By means of quadratizations, minimization of f is reduced to minimization (over its extended set of variables) of the quadratic functiong(x; y). This is of practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper [1] initiated a systematic study of the minimum number of auxiliary y-variables required in a quadratization of an arbitrary function f (a natural question, since the complexity of minimizing the quadratic function g(x; y) depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions f(x), those functions whose value depends only on the Hamming weight of the input x (the number of variables equal to 1).