A $\mu$-constrained Boolean Max-CSP$(\psi)$ instance is a Boolean Max-CSP instance on predicate $\psi:\{0,1\}^r \to \{0,1\}$ where the objective is to find a labeling of relative weight exactly $\mu$ that maximizes the fraction of satisfied constraints. In this work, we study the approximability of constrained Boolean Max-CSPs via SDP hierarchies by relating the integrality gap of Max-CSP $(\psi)$ to its $\mu$-dependent approximation curve. Formally, assuming the Small-Set Expansion Hypothesis, we show that it is NP-hard to approximate $\mu$-constrained instances of Max-CSP($\psi$) up to factor ${\sf Gap}_{\ell,\mu}(\psi)/\log(1/\mu)^2$ (ignoring factors depending on $r$) for any $\ell \geq \ell(\mu,r)$. Here, ${\sf Gap}_{\ell,\mu}(\psi)$ is the optimal integrality gap of $\ell$-round Lasserre relaxation for $\mu$-constrained Max-CSP($\psi$) instances. Our results are derived by combining the framework of Raghavendra [STOC 2008] along with more recent advances in rounding Lasserre relaxations and reductions from the Small-Set Expansion (SSE) problem. A crucial component of our reduction is a novel way of composing generic bias-dependent dictatorship tests with SSE, which could be of independent interest.

翻译：一个$\mu$-约束布尔Max-CSP$(\psi)$实例是指谓词$\psi:\{0,1\}^r \to \{0,1\}$上的布尔Max-CSP实例，其目标是在相对权重恰好为$\mu$的标记中最大化满足约束的比例。本文通过将Max-CSP $(\psi)$的积分间隙与其$\mu$依赖近似曲线相关联，研究利用SDP层次结构逼近约束布尔Max-CSP的可逼近性。形式上，假设小集扩张假设成立，我们证明：对于任意$\ell \geq \ell(\mu,r)$，将$\mu$-约束Max-CSP($\psi$)实例近似到因子${\sf Gap}_{\ell,\mu}(\psi)/\log(1/\mu)^2$（忽略与$r$相关的因子）是NP困难的。其中，${\sf Gap}_{\ell,\mu}(\psi)$是$\mu$-约束Max-CSP($\psi$)实例的$\ell$轮Lasserre松弛的最优积分间隙。我们的结果通过结合Raghavendra的框架[STOC 2008]与Lasserre松弛舍入和小集扩张问题归约的最新进展推导得出。归约的一个关键组成部分是创新性地将通用偏差依赖独裁测试与SSE结合，这一方法可能具有独立的研究价值。