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33
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 305 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
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 189 (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
An Optimal Lower Bound on the Number of Variables for Graph Identification
 Combinatorica
, 1992
"... In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open q ..."
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Cited by 135 (9 self)
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In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices. 1
The Graph Isomorphism Problem
, 1996
"... The graph isomorphism problem can be easily stated: check to see if two graphs that look differently are actually the same. The problem occupies a rare position in the world of complexity theory, it is clearly in NP but is not known to be in P and it is not known to be NPcomplete. Many subdiscipli ..."
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Cited by 63 (0 self)
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The graph isomorphism problem can be easily stated: check to see if two graphs that look differently are actually the same. The problem occupies a rare position in the world of complexity theory, it is clearly in NP but is not known to be in P and it is not known to be NPcomplete. Many subdisciplines of mathematics, such as topology theory and group theory, can be brought to bear on the problem, and yet only for special classes of graphs have polynomialtime algorithms been discovered. Incongruently, this problem seems very easy in practice. It is almost always trivial to check two random graphs for isomorphism, and fast hardware implementations exists for application domains such as image processing. This paper is mostly a survey of related work in the graph isomorphism field. We examine the problem from many angles, mirroring the multifaceted nature of the literature. We survey complexity results for the graph isomorphism problem, and discuss some of the classes of graphs which hav...
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 Hardness of Graph Isomorphism
 SIAM J. COMPUT
"... We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the stro ..."
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Cited by 33 (1 self)
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We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
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.
Generalpurpose mcmc inference over relational structures
 In Proceedings of the Proceedings of the TwentySecond Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI06
"... Tasks such as record linkage and multitarget tracking, which involve reconstructing the set of objects that underlie some observed data, are particularly challenging for probabilistic inference. Recent work has achieved efficient and accurate inference on such problems using Markov chain Monte Carl ..."
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Cited by 22 (6 self)
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Tasks such as record linkage and multitarget tracking, which involve reconstructing the set of objects that underlie some observed data, are particularly challenging for probabilistic inference. Recent work has achieved efficient and accurate inference on such problems using Markov chain Monte Carlo (MCMC) techniques with customized proposal distributions. Currently, implementing such a system requires coding MCMC state representations and acceptance probability calculations that are specific to a particular application. An alternative approach, which we pursue in this paper, is to use a generalpurpose probabilistic modeling language (such as BLOG) and a generic MetropolisHastings MCMC algorithm that supports usersupplied proposal distributions. Our algorithm gains flexibility by using MCMC states that are only partial descriptions of possible worlds; we provide conditions under which MCMC over partial worlds yields correct answers to queries. We also show how to use a contextspecific Bayes net to identify the factors in the acceptance probability that need to be computed for a given proposed move. Experimental results on a citation matching task show that our generalpurpose MCMC engine compares favorably with an applicationspecific system. 1
Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs
"... The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate ..."
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Cited by 15 (1 self)
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The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing stateoftheart tools.