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Nonmitotic sets
 Electronic Computational Complexity Colloquium
, 2006
"... We study the question of the existence of nonmitotic sets in NP. We show under various hypotheses that • 1ttmitoticity and mmitoticity differ on NP. • 1ttreducibility and mreducibility differ on NP. • There exist nonTautoreducible sets in NP (by a result from AmbosSpies, these sets are nei ..."
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We study the question of the existence of nonmitotic sets in NP. We show under various hypotheses that • 1ttmitoticity and mmitoticity differ on NP. • 1ttreducibility and mreducibility differ on NP. • There exist nonTautoreducible sets in NP (by a result from AmbosSpies, these sets are neither Tmitotic nor mmitotic). • Tautoreducibility and Tmitoticity differ on NP (this contrasts the situation in the recursion theoretic setting, where Ladner showed that autoreducibility and mitoticity coincide). • 2tt autoreducibility does not imply weak 2ttmitoticity. • 1ttcomplete sets for NP are nonuniformly mcomplete.
Logspace Mitoticity
, 2007
"... We study the relation of autoreducibility and mitoticity for polylogspace manyone reductions and logspace manyone reductions. For polylogspace these notions coincide, while proving the same for logspace is out of reach. More precisely, we show the following results with respect to nontrivial s ..."
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We study the relation of autoreducibility and mitoticity for polylogspace manyone reductions and logspace manyone reductions. For polylogspace these notions coincide, while proving the same for logspace is out of reach. More precisely, we show the following results with respect to nontrivial sets and manyone reductions. 1. polylogspace autoreducible ⇔ polylogspace mitotic 2. logspace mitotic ⇒ logspace autoreducible ⇒ (log n · log log n)space mitotic 3. relative to an oracle, logspace autoreducible � ⇒ logspace mitotic 1
The complexity of unions of disjoint sets
 In Proceedings 24th Symposium on Theoretical Aspects of Computer Science
, 2007
"... This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NPcomplete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we app ..."
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This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NPcomplete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of mequivalent disjoint sets. Under the hypothesis that NE � = coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets. 1
Mitosis in Computational Complexity
"... Abstract. This expository paper describes some of the results of two recent research papers [GOP + 05,GPSZ05]. The first of these papers proves that every NPcomplete set is manyone autoreducible. The second paper proves that every manyone autoreducible set is manyone mitotic. It follows immediat ..."
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Abstract. This expository paper describes some of the results of two recent research papers [GOP + 05,GPSZ05]. The first of these papers proves that every NPcomplete set is manyone autoreducible. The second paper proves that every manyone autoreducible set is manyone mitotic. It follows immediately that every NPcomplete set is manyone mitotic. Hence, we have the compelling result that every NPcomplete set A splits into two NPcomplete sets A1 and A2. 1 Autoreducibility We begin with the notion of autoreducibility. Traktenbrot [Tra70] defined a set A to be autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. This means there is an oracle Turing machine M such that A = L(M A) and M on input x never queries x. Ladner [Lad73] showed that there exist Turingcomplete recursively enumerable sets that are not autoreducible. We are interested in the polynomialtime variant of autoreducibility, introduced by AmbosSpies [AS84], where we require the oracle
The Fault Tolerance of NPHard Problems
"... Abstract. We study the effects of faulty data on NPhard sets. We consider hard sets for several polynomial time reductions, add corrupt data and then analyze whether the resulting sets are still hard for NP. We explain that our results are related to a weakened deterministic variant of the notion o ..."
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Abstract. We study the effects of faulty data on NPhard sets. We consider hard sets for several polynomial time reductions, add corrupt data and then analyze whether the resulting sets are still hard for NP. We explain that our results are related to a weakened deterministic variant of the notion of program selfcorrection by Blum, Luby, and Rubinfeld. Among other results, we use the LeftSet technique to prove that mcomplete sets for NP are nonadaptively weakly deterministically selfcorrectable while bttcomplete sets for NP are weakly deterministically selfcorrectable. Our results can also be applied to the study of Yesha’s pcloseness. In particular, we strengthen a result by Ogiwara and Fu. 1