Resolve P versus NP

Determine whether there exists a polynomial-time algorithm that solves all problems in NP; equivalently, prove or refute that any NP-complete problem admits a polynomial-time algorithm (i.e., resolve whether P equals NP).

Background

The paper motivates new optimization paradigms by emphasizing the prevalence and difficulty of NP-hard problems in real-world applications. It notes that many problems in NP reduce to NP-complete problems, so an efficient algorithm for any NP-complete problem would imply efficient algorithms for all NP problems.

Within this context, the authors explicitly acknowledge that the widely held belief that no polynomial-time algorithm exists for all problems in NP remains unproven. This foundational open problem underlies the significance of developing specialized heuristic and analog approaches such as the proposed entropy computing paradigm.

References

The computer science community largely believes that no polynomial time algorithm exists for all problems in NP (as an efficient algorithm for any of these complete problems, would imply), but this has yet to be proven.

— Entropy Computing, A Paradigm for Optimization in Open Photonic Systems (2407.04512 - Nguyen et al., 5 Jul 2024) in Introduction (Section 1)