Mahaney's theorem
From HandWiki
Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that if any sparse language is NP-complete, then P = NP. Also, if any sparse language is NP-complete with respect to Turing reductions, then the polynomial-time hierarchy collapses to [math]\displaystyle{ \Delta^P_2 }[/math].[1] Mahaney's argument does not actually require the sparse language to be in NP, so there is a sparse NP-hard set if and only if P = NP. This is because the existence of an NP-hard sparse set implies the existence of an NP-complete sparse set.[2]
References
- ↑ Mahaney, Stephen R. (October 1982). "Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis". Journal of Computer and System Sciences 25 (2): 130–143. doi:10.1016/0022-0000(82)90002-2.
- ↑ Balcázar, José Luis; Díaz, Josep; Gabarró, Joaquim (1990). Structural Complexity II. Springer. pp. 130–131. ISBN 3-540-52079-1.
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