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39
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 71 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
Counting Classes: Thresholds, Parity, Mods, and Fewness
, 1996
"... Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomialtime Turing machines. Well known examples of counting classes are NP, coNP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable ..."
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Cited by 61 (13 self)
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Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomialtime Turing machines. Well known examples of counting classes are NP, coNP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k iP = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [28]). 1. Introduction Valiant [27] defined the class #P of functions whose values equal the number of accepting paths of polynomialtime bo...
QueryLimited Reducibilities
, 1995
"... We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle. We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truthtable reduction) and queries made in serial (each question ..."
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Cited by 41 (14 self)
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We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle. We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truthtable reduction) and queries made in serial (each question being permitted to depend on the answers to the previous questions, as in a Turing reduction). We define computability by a set of functions, and we show that it captures the informationtheoretic aspects of computability by a fixed number of queries to an oracle. Using that concept, we prove a very powerful result, the Nonspeedup Theorem, which states that 2 n parallel queries to any fixed nonrecursive oracle cannot be answered by an algorithm that makes only n queries to any oracle whatsoever. This is the tightest general result possible. A corollary is the intuitively obvious, but nontrivial result that additional parallel queries to an oracle allow us to compute additional functions; t...
Computing Functions with Parallel Queries to NP
, 1993
"... The class \Theta p 2 of languages polynomialtime truthtable reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of \Theta p 2 based on these characterizations. We show that in this way the three function classes FL NP log , FP NP l ..."
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Cited by 39 (1 self)
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The class \Theta p 2 of languages polynomialtime truthtable reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of \Theta p 2 based on these characterizations. We show that in this way the three function classes FL NP log , FP NP log , and FP NP k are obtained. In contrast to the language case the function classes seem to all be different. We give evidence in support of this fact by showing that FL NP log coincides with any of the other classes then L = P, and that the equality of the classes FP NP log and FP NP k would imply that the number of nondeterministic bits needed for the computation of any problem in NP can be reduced by a polylogarithmic factor, and that the problem can be computed deterministically with a subexponential time bound of order 2 n O(1= log log n) . 1 Introduction The study of nondeterministic computation is a central topic in structural complexity theory. The acceptance mechanism of...
On the Power of NumberTheoretic Operations with Respect to Counting
 IN PROCEEDINGS 10TH STRUCTURE IN COMPLEXITY THEORY
, 1995
"... We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain t ..."
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Cited by 32 (9 self)
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We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized case. For closure properties f of #P, we have h#Pi f = #P. The other end of the range is marked by operations f for which h#Pi f corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that h#Pi f corresponds to some subclass C of the counting hierarchy. This will then imply that #P is closed under f if and only if ...
Relativized Counting Classes: Relations among Thresholds, Parity, and Mods
 Journal of Computer and System Sciences
, 1991
"... Well known complexity classes such as NP, coNP, \PhiP (PARITYP), and PP are produced by considering a nondeterministic polynomial time Turing machine N and defining acceptance in terms of the number of accepting paths in N . That is, they are subclasses of P #P[1] . Other interesting classes suc ..."
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Cited by 23 (5 self)
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Well known complexity classes such as NP, coNP, \PhiP (PARITYP), and PP are produced by considering a nondeterministic polynomial time Turing machine N and defining acceptance in terms of the number of accepting paths in N . That is, they are subclasses of P #P[1] . Other interesting classes such as MOD k P and C=P are also subclasses of P #P[1] . Many relations among these classes are unresolved. Of course, these classes coincide if P = PSPACE. However, we develop a simple combinatorial technique for constructing oracles that separate counting classes. Our results suggest that it will be difficult to resolve the unknown relationships among different counting classes. In addition to presenting new oracle separations, we simplify several previous constructions. 1. Introduction In [26], Valiant defined the class #P of counting functions. Definition 1 [Valiant] #P is the class of functions for which there exists a nondeterministic polynomialtime Turing machine N such that f(x) is...
The Boolean Isomorphism Problem
 SIAM JOURNAL ON COMPUTING
, 1996
"... We investigate the computational complexity of the Boolean Isomorphism problem (BI): on input of two Boolean formulas F and G decide whether there exists a permutation of the variables of G such that F and G become equivalent. Our main result is a oneround interactive proof for BI, where the verifi ..."
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Cited by 22 (2 self)
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We investigate the computational complexity of the Boolean Isomorphism problem (BI): on input of two Boolean formulas F and G decide whether there exists a permutation of the variables of G such that F and G become equivalent. Our main result is a oneround interactive proof for BI, where the verifier has access to an NP oracle. To obtain this, we use a recent result from learning theory by Bshouty et.al. that Boolean formulas can be learned probabilistically with equivalence queries and access to an NP oracle. As a consequence, BI cannot be \Sigma p 2 complete unless the Polynomial Hierarchy collapses. This solves an open problem posed in [BRS95]. Further properties of BI are shown: BI has And and Orfunctions, the counting version, #BI, can be computed in polynomial time relative to BI, and BI is selfreducible.
On the computational complexity of some classical equivalence relations on boolean functions
 Forschungsberichte Mathematische Logik, Universitat Heidelberg, Bericht Nr. 18, Dezember
, 1998
"... Abstract. The paper analyzes in terms of polynomial time manyone reductions the computational complexity of several natural equivalence relations on Boolean functions which derive from replacing variables by expressions, one of them is the Boolean isomorphism relation. Most of these computational p ..."
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Cited by 20 (4 self)
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Abstract. The paper analyzes in terms of polynomial time manyone reductions the computational complexity of several natural equivalence relations on Boolean functions which derive from replacing variables by expressions, one of them is the Boolean isomorphism relation. Most of these computational problems turn out to be between coNP and � p 2. 1.
Unambiguous Computation: Boolean Hierarchies and Sparse TuringComplete Sets
, 1994
"... This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turingcomplete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightfor ..."
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Cited by 19 (14 self)
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This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turingcomplete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightforwardly to UP. For example, it is known for NP (and more generally for any class containing \Sigma and ; and closed under union and intersection) that the symmetric difference hierarchy, the Boolean hierarchy, and the Boolean closure all are equal. We prove that closure under union is not needed for this claim: For any class K that contains \Sigma and ; and is closed under intersection (e.g., UP, US, and FewP), the symmetric difference hierarchy over K, the Boolean hierarchy over K, and the Boolean closure of K all are equal. On the other hand, we show that two hierarchiesthe Hausdorff hierarchy and the nested difference hierarchy which in the NP case are equal to the Boolean cl...