NP-Complete problems that admit an efficient algorithm under the promise of a unique solution
No NP-complete problem is known to admit a polynomial-time algorithm under uniqueness promise. Valiant and Vazirani theorem applies to any known natural NP-complete problem.
For all known NP-complete problems, there is a parsimonious reduction from 3SAT. Oded Goldreich states the fact that &quot;all known reductions among natural $NP$-complete problems are either parsimonious or can be easily modified to be so&quot;. ( Computational Complexity: A Conceptual Perspective By Oded Goldreich).
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