A $\mu$-biased Max-CSP instance with predicate $\psi:\{0,1\}^r \to \{0,1\}$ is an instance of Constraint Satisfaction Problem (CSP) where the objective is to find a labeling of relative weight at most $\mu$ 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 $\mu$ 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 $\mu$. 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 $k = \mu n$ is NP-hard to approximate to a factor of $\Omega(r^3\mu^{r-1}\log(1/\mu))$ for every $r \geq 2$ and $\mu < 2^{-r}$. We also give a $O(\mu^{r-1}\log(1/\mu))$-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.
翻译:以 $\ president $\ 0. 1\ r\ t 0. 1\ $ 美元为基底的 Max- CSP 实例 : @% 1, 1\ 美元 为 0. 1 美元 为 抑制性 CSP (CSP) 实例, 目标是在 $\ mu$ 最多找到相对重量的标签, 最多能满足限制的最大部分 。 错误的 CSP 是多功能的, 表示一些研究周密的问题, 如 Densest- $- Sub( Hyper) 和 Smm SetExplexion 。 在这项工作中, 我们探索偏差参数 $\ mum 美元( mutrial $ 美元) 的作用 。 在上调时, 我们的基调值是一个新的基值和基值 。 当我们使用硬值时, 我们的基值是每个基值的基值 。