Splitting NP-Complete Sets

We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long-standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al., complete sets for all of the following complexity classes are m-mitotic: $\mathrm{NP}$,...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 37; no. 5; pp. 1517 - 1535
Main Authors: Glaßer, Christian, Pavan, A., Selman, Alan L., Zhang, Liyu
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2008
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science
Computer science; control theory; systems
Exact sciences and technology
Logic and foundations
Mathematical logic, foundations, set theory
Mathematics
Miscellaneous
Recursion theory
Sciences and techniques of general use
Theoretical computing
Polynomial
68Q15
NP-complete sets
mitoticity; 68Q17
Complexity class
autoreducibility
Reducibility
Redundancy
Complexity
Polynomial time
computational and structural complexity
Splitting
Equivalence
Sparse set
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long-standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al., complete sets for all of the following complexity classes are m-mitotic: $\mathrm{NP}$, $\mathrm{coNP}$, $\oplus\mathrm{P}$, $\mathrm{PSPACE}$, and $\mathrm{NEXP}$, as well as all levels of $\mathrm{PH}$, $\mathrm{MODPH}$, and the Boolean hierarchy over $\mathrm{NP}$. In the cases of $\mathrm{NP}$, $\mathrm{PSPACE}$, $\mathrm{NEXP}$, and $\mathrm{PH}$, this at once answers several well-studied open questions. These results tell us that complete sets share a redundancy that was not known before. In particular, every $\mathrm{NP}$-complete set $A$ splits into two $\mathrm{NP}$-complete sets $A_1$ and $A_2$. We disprove the equivalence between autoreducibility and mitoticity for all polynomial-time-bounded reducibilities between 3-tt-reducibility and Turing-reducibility: There exists a sparse set in $\mathrm{EXP}$ that is polynomial-time 3-tt-autoreducible, but not weakly polynomial-time T-mitotic. In particular, polynomial-time T-autoreducibility does not imply polynomial-time weak T-mitoticity, which solves an open question by Buhrman and Torenvliet.
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ISSN:0097-5397
1095-7111
DOI:10.1137/060673886