Necessity of exponential time for optimal solutions to NP-hard optimization problems

Determine whether exponential time is strictly necessary in the worst case for algorithms that always find optimal solutions to NP-hard optimization problems, given the widely believed conjecture P ≠ NP and current evidence on super-polynomial complexity.

Background

In discussing the complexity landscape motivating quantum heuristics like QAOA, the authors reference the widely believed conjecture P ≠ NP and the lack of polynomial-time algorithms for NP-hard problems. They highlight a broader unresolved complexity-theoretic question about whether exponential time is strictly necessary in the worst case.

Resolving this would have deep implications not only for the feasibility of exact algorithms in quantized MIMO design but also for the broader field of combinatorial optimization.

References

NP-hard optimization problems admit no known polynomial-time solution; under the widely believed conjecture $\text{P}\neq\text{NP}$, any algorithm that always finds an optimal solution must run in super-polynomial (and typically exponential) time in the worst case, although it remains unproven whether exponential time is strictly necessary.

— Quantum Approximate Optimization Algorithm for MIMO with Quantized b-bit Beamforming (2510.15935 - Mitsiou et al., 7 Oct 2025) in Subsection "Motivation Contribution" (Section I)