We prove a PCP theorem for the existential theory of the reals, showing that MAX-ETR-INV is $\exists\mathbb{R}$-hard to approximate to within some constant factor. The existential theory of the reals (ETR) is a decision problem asking if there exists a set of real-valued variables satisfying some constraints involving polynomials and inequalities, and $\exists\mathbb{R}$ is the complexity class of problems polynomial-time reducible to ETR. Many important geometric problems are known to be $\exists\mathbb{R}$-complete. $\exists\mathbb{R}$-hardness results frequently work by a reduction from the $\exists\mathbb{R}$-complete problem ETR-INV, which asks if there is a an assignment of real variables each in the interval $[\frac12, 2]$ satisfying some constraints of form $x=1$, $xy=1$ and $x+y=z$. MAX-ETR-INV is a related optimization problem that asks, given a set of constraints of form $x=1$, $xy=1$, and $x+y=z$, for a feasible (that is, satisfiable with variables in $[\frac12, 2]$) subset of those constraints of the largest possible size. We show that there is some constant $ε>0$ such that it is $\exists\mathbb{R}$-hard to approximate MAX-ETR-INV better than a $1-ε$ factor. This means that even a non-deterministic polynomial-time algorithm can't approximate MAX-ETR-INV better than this factor unless $\exists\mathbb{R}=\text{NP}$. We also give a polynomial-time $8$-factor approximation algorithm and a non-deterministic-polynomial-time $2$-factor approximation algorithm for MAX-ETR-INV.

翻译：我们证明了关于实数存在理论的一个PCP定理，表明在一定常数因子内逼近MAX-ETR-INV问题是$\exists\mathbb{R}$-难的。实数存在理论（ETR）是一类判定问题，询问是否存在一组实数值变量满足包含多项式与不等式的某些约束，而$\exists\mathbb{R}$是可在多项式时间内归约至ETR的问题的复杂度类。许多重要的几何问题已知是$\exists\mathbb{R}$-完全的。$\exists\mathbb{R}$-难的结果常通过从$\exists\mathbb{R}$-完全问题ETR-INV进行归约得到，该问题询问是否存在每个变量取值区间为$[\frac12, 2]$的实数赋值，满足形如$x=1$、$xy=1$和$x+y=z$的约束。MAX-ETR-INV是一个相关的优化问题，给定一组形如$x=1$、$xy=1$和$x+y=z$的约束，要求找出其中最大可能大小的可行子集（即变量在$[\frac12, 2]$内可满足的约束子集）。我们证明存在某个常数$ε>0$，使得以优于$1-ε$因子的精度逼近MAX-ETR-INV是$\exists\mathbb{R}$-难的。这意味着除非$\exists\mathbb{R}=\text{NP}$，否则即使非确定性多项式时间算法也无法以优于该因子的精度逼近MAX-ETR-INV。我们还给出了MAX-ETR-INV的一个多项式时间$8$因子近似算法和一个非确定性多项式时间$2$因子近似算法。