Anthony, Martin and Boros, Endre and Crama, Yves and Gruber, Aritanan
(2013)
Quadratization of symmetric pseudoBoolean functions.
RUTCOR Research Reports,
RRR 122013.
Rutgers University, Rutgers Center for Operations Research, New Jersey, USA.
Abstract
A pseudoBoolean function is a realvalued function f(x)=f(x_1,x_2,...,x_n) of n binary variables; that is, a mapping from {0,1}^n to the real numbers. For a pseudoBoolean function f(x) on {0,1}^n, 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 binary variables y_1,y_2,...,y_m such that f(x)= min {g(x,y) : y in {0,1}^m} for all x in {0,1}^n. By means of quadratizations, minimization of f is reduced to minimization (over its extended set of variables) of the quadratic function g(x,y). This is of some 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 (by the current authors) initiated a systematic study of the minimum number of auxiliary yvariables 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 pseudoBoolean functions f(x), those functions whose value depends only on the Hamming weight of the input x (the number of variables equal to 1).
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