NP-Completeness of Deterministic Communication Complexity via Relaxed Interlacing

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.