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45
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
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 ...
Quantum algorithms: Entanglement enhanced information processing
 Phil. Trans. R. Soc. Lond. A
, 1998
"... Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algori ..."
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Cited by 31 (1 self)
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Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2 n in the quantum context, showing how the grouptheoretic formalism leads to the standard quantum network and identifying the property of entanglement that gives rise to the exponential speedup (compared to the classical FFT). Finally we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.
Entanglement and quantum computation
 The Geometric Universe
, 1998
"... The phenomenon of quantum entanglement is perhaps the most enigmatic feature of the formalism of quantum theory. It underlies many of the most curious and controversial aspects of the quantum mechanical description of the world. In [1] Penrose gives a delightful and accessible account of entanglemen ..."
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Cited by 30 (3 self)
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The phenomenon of quantum entanglement is perhaps the most enigmatic feature of the formalism of quantum theory. It underlies many of the most curious and controversial aspects of the quantum mechanical description of the world. In [1] Penrose gives a delightful and accessible account of entanglement illustrated by some
On Approximation Algorithms for the Minimum Satisfiability Problem
 Information Processing Letters
, 1996
"... this paper, our focus is on deterministic approximation algorithms for the MINSAT problem. From now on, we will use the word `heuristic' to mean a deterministic approximation algorithm which runs in polynomial time. Note that when the clauses are of size \Theta(n), where n is the number of vari ..."
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Cited by 25 (0 self)
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this paper, our focus is on deterministic approximation algorithms for the MINSAT problem. From now on, we will use the word `heuristic' to mean a deterministic approximation algorithm which runs in polynomial time. Note that when the clauses are of size \Theta(n), where n is the number of variables, the heuristic analyzed in [18] provides only a weak performance guarantee of \Theta(n). We present a simple approximationpreserving reduction from MINSAT to the minimum vertex cover (MINVC) problem. This reduction, in conjunction with known heuristics for the MINVC problem (see for example, [8,22]), yields a heuristic with a performance guarantee of 2 for MINSAT, thus improving the result of Kohli et al. [18]. We also show that MINSAT is as hard to approximate as MINVC; that is, if there is a heuristic with a performance guarantee ae for MINSAT, then there is a heuristic with the same performance guarantee ae for MINVC. Moreover, we show that this result holds even for MINSAT instances defined by Horn formulas. It has been conjectured in [12] that no polynomial approximation algorithm can provide a performance guarantee of 2 \Gamma ffl for any fixed ffl ? 0 for MINVC unless P = NP. Thus, our result provides an indication of the difficulty involved in devising a heuristic with a performance guarantee better than 2 for MINSAT.
Unrestricted vs restricted cut in a tableau method for Boolean circuits
 In: AI&M 2004, 8th International Symposium on Artificial Intelligence and Mathematics
, 2005
"... This paper studies the relative proof complexity of variations of a tableau method for Boolean circuit satisfiability checking obtained by restricting the use of the cut rule in several natural ways. The results show that the unrestricted cut rule can be exponentially more effective than any of th ..."
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Cited by 23 (4 self)
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This paper studies the relative proof complexity of variations of a tableau method for Boolean circuit satisfiability checking obtained by restricting the use of the cut rule in several natural ways. The results show that the unrestricted cut rule can be exponentially more effective than any of the considered restrictions. Moreover, there are exponential differences between the restricted versions, too. The results also apply to the DavisPutnam procedure for conjunctive normal form formulae obtained from Boolean circuits with a standard linear size translation.
On the Complexity of Constructing Evolutionary Trees
, 1999
"... In this paper we study a few important tree optimization problems with applications to computational biology. These problems ask for trees that are consistent with an as large part of the given data as possible. We show that the maximum homeomorphic agreement subtree problem cannot be approximated w ..."
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Cited by 18 (8 self)
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In this paper we study a few important tree optimization problems with applications to computational biology. These problems ask for trees that are consistent with an as large part of the given data as possible. We show that the maximum homeomorphic agreement subtree problem cannot be approximated within a factor of N ffl , where N is the input size, for any 0 ffl ! 1 9 in polynomial time unless P=NP, even if all the given trees are of height 2. On the other hand, we present an O(N log N)time heuristic for the restriction of this problem to instances with O(1) trees of height O(1) yielding solutions within a constant factor of the optimum. We prove that the maximum inferred consensus tree problem is NPcomplete, and provide a simple, fast heuristic for it yielding solutions within one third of the optimum. We also present a more specialized polynomialtime heuristic for the maximum inferred local consensus tree problem.
The achievable region method in the optimal control of queueing systems; formulations, bounds and policies
 QUEUEING SYST
, 1995
"... We survey a new approach that the author and his coworkers have developed to formulate stochastic control problems (predominantly queueing systems) as mathematicalprogramming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., fi ..."
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Cited by 15 (4 self)
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We survey a new approach that the author and his coworkers have developed to formulate stochastic control problems (predominantly queueing systems) as mathematicalprogramming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable systems (multiclass single station queues and multiarmed bandit problems), restless bandit problems, polling systems, multiclass queueing and loss networks. These relaxations lead to bounds on the performance of an optimal policy. Using information from the relaxations we construct heuristic nearly optimal policies, The theme in the paper is the thesis that better formulations lead to deeper understanding and better solution methods. Overall the proposed approach for stochastic control problems parallels efforts of the mathematical programming community in the last twenty years to develop sharper formulations (polyhedral combinatorics and more recently nonlinear relaxations) and leads to new insights ranging from a complete characterization and new algorithms for indexable systems to tight lower bounds and nearly optimal algorithms for restless bandit problems, polling systems, multiclass queueing and loss networks.