# Quadratization of symmetric pseudo-Boolean functions

Anthony, Martin, Boros, Endre, Crama, Yves and Gruber, Aritanan (2013) Quadratization of symmetric pseudo-Boolean functions. RUTCOR Research Reports (RRR 12-2013). Rutgers University, Rutgers Center for Operations Research, New Jersey, USA.

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## Abstract

A pseudo-Boolean function is a real-valued 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 pseudo-Boolean 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 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).

Item Type: Monograph (Report) http://rutcor.rutgers.edu/ © 2013 The Authors Mathematics Q Science > QA Mathematics 03 Dec 2014 15:58 21 Aug 2021 23:06 http://eprints.lse.ac.uk/id/eprint/54799