We prove that computing the deterministic communication complexity of a Boolean function, given its truth table, is \textsf{NP}-complete in the standard protocol-tree-depth model, addressing a meta-complexity question raised by Yao in 1979. The reduction is from \(\{0,1\}\)-Vector Bin Packing and produces, in polynomial time, a communication matrix whose optimal protocol depth exhibits a one-bit gap between satisfiable and unsatisfiable instances. The main technical contribution is the \emph{relaxed-interlacing} framework that makes this reduction possible. It replaces exponential-size Cartesian products with polynomial-size almost \(t\)-wise independent column sets, a pseudorandom substitute for full products, while preserving the lower-bound and protocol-control statements needed for the reduction. We develop these statements in two stages: first for classical interlacing, where projection arguments give clean lower bounds and separation statements, and then for relaxed interlacing, where a bridge lemma recovers the classical lower-bound and separation statements with controlled density loss. This leads to an extension theorem that lifts the classical lower bound to the relaxed setting and a near-exact separation theorem that lifts the corresponding protocol-control statement, with the present \textsf{NP}-completeness theorem as their main application here.

翻译：我们证明：在标准协议树深度模型下，给定布尔函数的真值表，判定其确定性通信复杂度是\textsf{NP}-完全的，解决了Yao于1979年提出的元复杂度问题。该归约来自\(\{0,1\}\)向量装箱问题，并在多项式时间内生成一个通信矩阵，其最优协议深度在可满足与不可满足实例间呈现单比特间隙。主要技术贡献在于提出了实现此归约的\emph{松弛交织}框架。该框架使用多项式大小的近似\(t\)阶独立列集（作为全乘积的伪随机替代）替代指数规模的笛卡尔积，同时保留归约所需的下界与控制协议特性。我们分两阶段建立这些特性：首先针对经典交织，利用投影论证获得清晰下界与分离陈述；随后针对松弛交织，通过桥接引理以受控密度损失恢复经典下界与分离陈述。由此得到将经典下界提升至松弛环境的扩展定理，以及提升相应协议控制特性的近精确分离定理——本文的核心应用即在于此\textsf{NP}-完全性定理。