Reference : Quadratization of symmetric pseudo-Boolean functions
Scientific journals : Article
Business & economic sciences : Quantitative methods in economics & management
Physical, chemical, mathematical & earth Sciences : Mathematics
Quadratization of symmetric pseudo-Boolean functions
Anthony, Martin [London School of Economics > > > >]
Boros, Endre [Rutgers University (New Jersey) - RU > > > >]
Crama, Yves mailto [Université de Liège - ULiège > HEC-Ecole de gestion : UER > Recherche opérationnelle et gestion de la production >]
Gruber, Aritanan [Rutgers University (New Jersey) - RU > > > >]
Discrete Applied Mathematics
Elsevier Science
Yes (verified by ORBi)
The Netherlands
[en] pseudo-Boolean function ; quadratic optimization ; symmetric function ; nonlinear integer programming
[en] A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to ${\bbr}$. 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 $y_1,y_2,\ldots,y_m$ 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 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 \emph{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).
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