Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is ${NP}$-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.

In this work, we establish the first ${NP}$-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function $f \colon \{0,1\}^n \to \{0,1\}^m$ is ${NP}$-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler ${NP}$-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.

Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are ${NP}$-hard under deterministic reductions. In particular, unless ${P} = {NP}$, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.