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15
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
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...
The Isomorphism Conjecture Fails Relative to a Random Oracle
 J. ACM
, 1996
"... Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kc ..."
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Cited by 40 (4 self)
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Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kcreative setsand defined a class of sets (the K k f 's) that are necessarily kcreative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NPcomplete sets. Clearly, the BermanHartmanis and JosephYoung conjectures cannot both be correct. We introduce a family of strong oneway functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NPcomplete sets, as Joseph and Young conjectured, and the BermanHartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...
NPhard Sets are PSuperterse Unless R = NP
, 1992
"... A set A is pterse (psuperterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of ..."
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Cited by 27 (5 self)
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A set A is pterse (psuperterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of functions reducible to A via a polynomialtime Turing reduction that makes at most q queries. A set A is pterse if PF A qtt 6` PF A (q\Gamma1)T for all constants q. A is psuperterse if PF A qtt 6` PF X qT for all constants q and sets X . We show that all NPhard sets (under p tt reductions) are psuperterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NPcomplete sets are psuperterse unless P = UP (oneway functions fail to exist), R = NP (there exist randomized polynomialtime algorithms for all problems in NP), and the polynomialtime hierarchy collapses. This mostly solves the main open...
ComplexityTheoretic Aspects of Interactive Proof Systems
, 1989
"... In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have a ..."
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Cited by 19 (3 self)
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In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zeroknowledge interactive protocol convinces a verifier that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zeroknowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zeroknowledge protocols for NPcomplete languages unless the polynomialtime hierarchy collapses. Thus knowledge comp...
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.
Efficiently Approximable RealValued Functions
 Electronic Colloquium on Computational Complexity
, 2000
"... We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a nat ..."
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Cited by 12 (2 self)
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We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a natural complete problem: computing the acceptance probability of a given Boolean circuit; in contrast, no complete problems are known for BPP. We observe that all known complexitytheoretic assumptions under which BPP is easy (i.e., can be efficiently derandomized) imply that APP is easy; on the other hand, we show that BPP may be easy while APP is not, by constructing an appropriate oracle. 1 Introduction The complexity class BPP is traditionally considered a class of languages that can be efficiently decided with the help of randomness. While it does contain some natural problems, the "semantic" nature of its definition (on every input, a BPP machine must have either at least 3=4 or at...
A General Method to Construct Oracles Realizing Given Relationships between Complexity Classes
, 1994
"... We present a method to prove oracle theorems of the following type. ..."
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Cited by 9 (1 self)
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We present a method to prove oracle theorems of the following type.
A Tight Relationship between Generic Oracles and Type2 Complexity Theory
, 1997
"... We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct. ..."
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Cited by 6 (1 self)
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We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct.
Simultaneous Strong Separations of Probabilistic and Unambiguous Complexity Classes
, 1992
"... We study the relationship between probabilistic and unambiguous computation, and provide strong relativized evidence that they are incomparable. In particular, we display a relativized world in which the complexity classes embodying these paradigms of computation are mutually immune. We answer q ..."
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Cited by 4 (0 self)
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We study the relationship between probabilistic and unambiguous computation, and provide strong relativized evidence that they are incomparable. In particular, we display a relativized world in which the complexity classes embodying these paradigms of computation are mutually immune. We answer questions formulated inand extend the line of research opened byGeske and Grollman [15] and Balcazar and Russo [3]. 1 Introduction: Why Compare Computational Paradigms? Many complexity classes have been defined in recent years to characterize the computational powers of natural approaches to computation. However, # Some of these results were announced at the 1989 International Conference on Computing and Information, Toronto, Canada. + Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, CA 94304. Research performed in part while at Columbia University, supported in part by NSF grants DCR8511713 and CCR8605353. # Department of Computer Science, University of Roc...