Exponential-time necessity for exact solutions to NP-hard optimization problems

Ascertain whether exponential time is strictly necessary in the worst case for algorithms that always produce exact optimal solutions to NP-hard optimization problems, even under the assumption that P ≠ NP.

Background

In discussing the computational complexity of the quantized beamforming problem, the authors situate it within the broader landscape of NP-hard optimization. They note the common belief that exact algorithms for NP-hard problems require super-polynomial time in the worst case under P ≠ NP.

They explicitly point out that, despite this understanding, it remains unproven whether exponential time is strictly necessary, highlighting a fundamental unresolved question in complexity theory relevant to the paper’s problem context.