What does NP-completeness signify in decision problems?

NP-completeness signifies that a problem is among the hardest problems in the class of NP (nondeterministic polynomial time). Specifically, if a decision problem is NP-complete, it means that there is no known polynomial-time algorithm to solve that problem, although a solution can be verified in polynomial time if one is provided.

This is critical in computer science because, although no efficient algorithm has been found for NP-complete problems, they have been shown to be solvable in non-polynomial time, and if a polynomial-time algorithm were discovered for any NP-complete problem, it would imply that all problems in NP can be solved in polynomial time—this is the essence of the famous P vs. NP problem.

The option indicating that NP-complete problems are incapable of being solved at all is misleading as it suggests impossibility rather than the current status of algorithmic efficiency. Similarly, the idea that NP-complete problems require exponential time for all algorithms is not accurate, as there are algorithms that can solve NP-complete problems (although they don't do so in polynomial time). The assertion that NP-complete problems can be solved in polynomial time is directly contradictory to the understanding of NP-completeness. Thus, the correct interpretation aligns accurately with