对称伪布尔函数的平方化
Quadratization of Symmetric Pseudo-Boolean Functions
摘要：
伪布尔函数是$n$二进制变量的实值函数$f(x)=f(x_1,x_2,\ldots,x_n)$;即，从$\{0,1\}^n$到$\mathbb{R}$的映射。对于$\{0,1\}^n$上的伪布尔函数$f(x)$，我们说$g(x,y)$是$f$的二次化，如果$g(x,y)$是依赖于$x$和$m$辅助二进制变量$y_1,y_2,\ldots,y_m$的二次多项式，使得对于所有$x \in \{0,1\}^n$，$f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$。通过二次化，$f$的最小化被简化为二次函数$g(x,y)$的最小化（在其扩展变量集上）。这是一些实际的利益，因为二次函数的最小化已经彻底研究了过去几十年，并取得了很大的进展，在解决这些问题的准确或启发式。一个相关的文件\cite{ABCG}发起了一个系统的研究的最小数目的辅助$y$ -变量所需的二次化的任意函数$f$（一个自然的问题，因为复杂性最小化的二次函数$g(x,y)$取决于，除其他因素外，对一些二进制变量）。在本文中，我们更精确地确定所需的辅助变量的数量的对称伪布尔函数$f(x)$，这些功能的值只取决于汉明重量的输入$x$（变量的数量等于$1$）的平方。
英文摘要：
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 $\mathbb{R}$. 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 \cite{ABCG} 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 $1$).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
