有偏CSP逼近性的一个特征
A Characterization of Approximability for Biased CSPs
摘要：
带有谓词$ψ:\{0,1\}^r \to \{0,1\}$的$μ$偏置Max-CSP实例是约束满足问题（CSP）的实例，其中目标是找到满足约束的最大分数的相对权重至多为$μ$的标记。有偏的CSP是通用的，并表示几个研究得很好的问题，如Denmark- $k$ -Sub（Hyper）图和SmallSetExpansion。在这项工作中，我们探讨了偏置参数$μ$对有偏CSP的可逼近性所起的作用。我们表明，这种CSP的可逼近性可以使用Denmark- $k$ -SubHypergraph（DkSH）的偏置逼近曲线来表征（直到损失元数因子$r$）。特别是，这给出了一个严格的特征的谓词承认近似保证是独立的偏置参数$μ$。基于上述动机，我们给出了DkSH的新近似和硬度结果。特别地，在小集合扩展假设（SSEH）下，证明了对于每个$r \geq 2$和$μ< 2^{-r}$，具有元$r$和$k = μn$的DkSH都是NP-难逼近因子$Ω(r^3μ^{r-1}\log(1/μ))$的.我们也给出了一个$O(μ^{r-1}\log(1/μ))$ -近似算法相同的设置。我们的上限和下限是紧的常数的因素，当arity $r$是一个常数，特别是，意味着第一紧近似界的Denmark- $k$ -子图问题的线性偏置制度。此外，利用上述特征，我们的结果也意味着匹配算法和硬度为每一个偏置CSP的常数。
英文摘要：
A $μ$-biased Max-CSP instance with predicate $ψ:\{0,1\}^r \to \{0,1\}$ is an instance of Constraint Satisfaction Problem (CSP) where the objective is to find a labeling of relative weight at most $μ$ which satisfies the maximum fraction of constraints. Biased CSPs are versatile and express several well studied problems such as Densest-$k$-Sub(Hyper)graph and SmallSetExpansion. In this work, we explore the role played by the bias parameter $μ$ on the approximability of biased CSPs. We show that the approximability of such CSPs can be characterized (up to loss of factors of arity $r$) using the bias-approximation curve of Densest-$k$-SubHypergraph (DkSH). In particular, this gives a tight characterization of predicates which admit approximation guarantees that are independent of the bias parameter $μ$. Motivated by the above, we give new approximation and hardness results for DkSH. In particular, assuming the Small Set Expansion Hypothesis (SSEH), we show that DkSH with arity $r$ and $k = μn$ is NP-hard to approximate to a factor of $Ω(r^3μ^{r-1}\log(1/μ))$ for every $r \geq 2$ and $μ< 2^{-r}$. We also give a $O(μ^{r-1}\log(1/μ))$-approximation algorithm for the same setting. Our upper and lower bounds are tight up to constant factors, when the arity $r$ is a constant, and in particular, imply the first tight approximation bounds for the Densest-$k$-Subgraph problem in the linear bias regime. Furthermore, using the above characterization, our results also imply matching algorithms and hardness for every biased CSP of constant arity.
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。
