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A Taxonomy of Complexity Classes of Functions
 Journal of Computer and System Sciences
, 1992
"... This paper comprises a systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP. There are three components to this work. ffl A taxonomy is presented that demonstrates all known inclusion relations of these cla ..."
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Cited by 88 (12 self)
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This paper comprises a systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP. There are three components to this work. ffl A taxonomy is presented that demonstrates all known inclusion relations of these classes. For (nearly) each inclusion that is not shown to hold, evidence is presented to indicate that the inclusion is false. As an example, consider FewPF, the class of multivalued functions that are nondeterministically computable in polynomial time such that for each x, there is a polynomial bound on the number of distinct output values of f(x). We show that FewPF ` PF NP tt . However, we show PF NP tt ` FewPF if and only if NP = coNP, and thus PF NP tt ` FewPF is likely to be false. ffl Whereas it is known that P NP (O(log n)) = P NP tt ` P NP [Hem87, Wagb, BH88], we show that PF NP (O(log n)) = PF NP tt implies P = FewP and R = NP. Also, we show that PF NP tt = PF ...
PSelective Sets, and Reducing Search to Decision vs. SelfReducibility
, 1993
"... We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to de ..."
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Cited by 39 (9 self)
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We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not selfreducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992Dec. 1992. Current address: Department of Computer Science, University of ElectroCommunications, Chofushi, Tokyo 182, Japan.  Department of Computer Science, State University of New York at Buffalo, 226...
The complexity of decision versus search
 SIAM Journal on Computing
, 1994
"... A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not red ..."
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Cited by 33 (1 self)
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A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NPcompleteness, selfreducibility, interactive proofs, program checking, sparse sets,
Easy sets and hard certificate schemes
 Acta Informatica
, 1997
"... Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for al ..."
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Cited by 16 (4 self)
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Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. In particular, we provide equivalent characterizations of these classes in terms of relative generalized Kolmogorov complexity, showing that they are robust. We also provide structural conditions—regarding immunity and class collapses—that put upper and lower bounds on the sizes of these two classes. Finally, we provide negative results showing that some of our positive claims are optimal with regard to being relativizable. Our negative results are proven using a novel observation: we show that the classical “wide spacing ” oracle construction technique yields instant nonbiimmunity results. Furthermore, we establish a result that improves upon Baker, Gill, and Solovay’s classical result that NP = P = NP ∩ coNP holds in some relativized world.
Molecular Computing, Bounded Nondeterminism, and Efficient Recursion
 In Proceedings of the 24th International Colloquium on Automata, Languages, and Programming
, 1998
"... The maximum number of strands used is an important measure of a molecular algorithm's complexity. This measure is also called the volume used by the algorithm. Every problem that can be solved by an NP Turing machine with b(n) binary nondeterministic choices can be solved by molecular computati ..."
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Cited by 14 (5 self)
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The maximum number of strands used is an important measure of a molecular algorithm's complexity. This measure is also called the volume used by the algorithm. Every problem that can be solved by an NP Turing machine with b(n) binary nondeterministic choices can be solved by molecular computation in a polynomial number of steps, with four test tubes, in volume 2 b(n) . We identify a large class of recursive algorithms that can be implemented using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3SAT, independent set, and 3colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 20]. The models we define were inspired as well by the work of [3, 28]. A molecular sequence is a string over an alphabet \Sigma (we can use any alphabet we like, encoding characters of \Sigma by finite sequences of base pairs). A test tube is a multiset of molecular sequences. We describe the allowable operations below. Whe...
Properties of NPcomplete sets
 In Proceedings of the 19th IEEE Conference on Computational Complexity
, 2004
"... We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreo ..."
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Cited by 10 (7 self)
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We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p mhard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences for longstanding open questions about whether NPcomplete sets are immune. For example, assuming that pseudorandom generators and secure oneway permutations exist, it follows easily that NPcomplete sets are not pimmune. Assuming only that secure oneway permutations exist, we prove that no NPcomplete set is DTIME(2nɛ)immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomialclose to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for NP are not closed under union. Our hypothesis asserts existence of a UPmachine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ timebounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)biimmune sets, then this hypothesis is true.
Biimmunity Results for Cheatable Sets
 Theoretical Computer Science
, 1995
"... An oracle A is kcheatable if there is a polynomialtime algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1cheatable sets cannot be biimmune for P. In contrast, we construct 2cheatable sets that are biim ..."
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Cited by 10 (6 self)
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An oracle A is kcheatable if there is a polynomialtime algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1cheatable sets cannot be biimmune for P. In contrast, we construct 2cheatable sets that are biimmune for arbitrary time complexity classes. In addition, for each k, we construct a set that is (k + 1)cheatable, but not kcheatable; we show that this separation does not hold with biimmunity. We show that if a recursive set A is biimmune for P then there exists an infinite 1cheatable set that is polynomialtime mreducible to A. Consequently if NP contains a set that is biimmune for P then NP contains a set that is not polynomialtime Turingequivalent to a selfreducible set. 1. Introduction Complexity theory deals with how hard problems are. Time, space, and alternation have served as measures of difficulty. Recently, researchers have Research supported by a Fannie and John Hertz ...
On Quasilinear Time Complexity Theory
, 1994
"... This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomialtime hierarchy carry over to the quasilineartime hierarchy. ..."
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Cited by 3 (0 self)
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This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomialtime hierarchy carry over to the quasilineartime hierarchy.
Computing complete graph isomorphisms and hamiltonian cycles from partial ones
 THEORY OF COMPUTING SYSTEMS
, 2002
"... We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism ϕ between two isomorphic graphs is as hard as computing ϕ itself. This result optimally improves upon a result of Gál, Halevi, Lipton, and Petrank. We establish a similar, albeit slightly weaker, resu ..."
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Cited by 2 (1 self)
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We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism ϕ between two isomorphic graphs is as hard as computing ϕ itself. This result optimally improves upon a result of Gál, Halevi, Lipton, and Petrank. We establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.
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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Cited by 2 (2 self)
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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