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Local Majority Voting, Small Coalitions and Controlling Monopolies in Graphs: A Review
 IN PROC. OF 3RD COLLOQUIUM ON STRUCTURAL INFORMATION AND COMMUNICATION COMPLEXITY
, 1996
"... This paper provides an overview of recent developments concerning the process of local majority voting in graphs, and its basic properties, from graph theoretic and algorithmic standpoints. ..."
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Cited by 30 (1 self)
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This paper provides an overview of recent developments concerning the process of local majority voting in graphs, and its basic properties, from graph theoretic and algorithmic standpoints.
The power of small coalitions in graphs
 In Proc. of 2nd Colloquium on Structural Information and Communication Complexity
, 1995
"... This paper considers the question of the in uence of a coalition of vertices, seeking to gain control (or majority) in local neighborhoods in a general graph. Say that a vertex v is controlled by the coalition M if the majority of its neighbors are from M. We ask how many vertices (as a function of ..."
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Cited by 12 (2 self)
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This paper considers the question of the in uence of a coalition of vertices, seeking to gain control (or majority) in local neighborhoods in a general graph. Say that a vertex v is controlled by the coalition M if the majority of its neighbors are from M. We ask how many vertices (as a function of jM j) canM control in this fashion. Upper and lower bounds are provided for this problem, as well as for cases where the majority is computed over larger neighborhoods (either neighborhoods of some xed radius r 1, or all neighborhoods of radii up to r). In particular, we look also at the case where the coalition must control all vertices outside itself, and derive bounds for its size.
The Computational Power of Discrete Hopfield Nets with Hidden Units
 Neural Computation
, 1996
"... We prove that polynomial size discrete Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks wi ..."
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Cited by 11 (6 self)
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We prove that polynomial size discrete Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly, i.e., the class computed by polynomial timebounded nonuniform Turing machines.
Tight Bounds on the size of 2monopolies
, 1996
"... This paper deals with the question of the influence of a monopoly of vertices, seeking to gain the majority in local neighborhoods in a graph. Say that a vertex v is r controlled by a set of vertices M if the majority of its neighbors at distance r are from M . We ask how large must M be in orde ..."
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Cited by 11 (2 self)
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This paper deals with the question of the influence of a monopoly of vertices, seeking to gain the majority in local neighborhoods in a graph. Say that a vertex v is r controlled by a set of vertices M if the majority of its neighbors at distance r are from M . We ask how large must M be in order to rmonopolize the graph, namely, rcontrol every vertex. Tight upper and lower bounds are provided for this problem, establishing that in an nvertex graph, an rmonopoly M (for any even r 2) must be of size \Omega\Gamma n 3=5 ), and that for any r 2 there exist nvertex graphs with rmonopolies of size O(n 3=5 ). This settles a problem left open in [LPRS93, BePe95].
An Overview Of The Computational Power Of Recurrent Neural Networks
 Proceedings of the 9th Finnish AI Conference STeP 2000{Millennium of AI, Espoo, Finland (Vol. 3: "AI of Tomorrow": Symposium on Theory, Finnish AI Society
, 2000
"... INTRODUCTION The two main streams of neural networks research consider neural networks either as a powerful family of nonlinear statistical models, to be used in for example pattern recognition applications [6], or as formal models to help develop a computational understanding of the brain [10]. His ..."
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Cited by 10 (3 self)
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INTRODUCTION The two main streams of neural networks research consider neural networks either as a powerful family of nonlinear statistical models, to be used in for example pattern recognition applications [6], or as formal models to help develop a computational understanding of the brain [10]. Historically, the brain theory interest was primary [32], but with the advances in computer technology, the application potential of the statistical modeling techniques has shifted the balance. 1 The study of neural networks as general computational devices does not strictly follow this division of interests: rather, it provides a general framework outlining the limitations and possibilities aecting both research domains. The prime historic example here is obviously Minsky's and Papert's 1969 study of the computational limitations of singlelayer perceptrons [34], which was a major inuence in turning away interest from neural network learning to symbolic AI techniques for more
On the Computational Power of Discrete Hopfield Nets
 In: Proc. 20th International Colloquium on Automata, Languages, and Programming
, 1993
"... . We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also th ..."
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Cited by 7 (4 self)
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. We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial spacebounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly. 1 Background Recurrent, or cyclic, neural networks are an intriguing model of massively parallel computation. In the recent surge of research in neural computation, such networks have been considered mostly from the point of view of two types of applications: pattern classification and associative memory (e.g. [16, 18, 21, 24]), and combinatorial optimization (e.g. [1, 7, 20]). Nevertheless, recurrent networks are capable also of more general types of computation, and issues of what exactly such networks can compute, and how they should be programmed, are becoming increasingly topica...
Group Updates and Multiscaling: An Efficient Neural Network Approach to Combinatorial Optimization
 IEEE Transactions on Systems, Man, and Cybernetics  Part B: Cybernetics
, 1996
"... A multiscale method is described in the context of binary Hopfieldtype neural networks. The appropriateness of the proposed technique for solving several classes of optimization problems is established by means of the notion of group update which is introduced here and investigated in relation to ..."
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Cited by 4 (4 self)
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A multiscale method is described in the context of binary Hopfieldtype neural networks. The appropriateness of the proposed technique for solving several classes of optimization problems is established by means of the notion of group update which is introduced here and investigated in relation to the properties of multiscaling. The method has been tested in the solution of partitioning and covering problems, for which an original mapping to Hopfieldtype neural networks has been developed. Experimental results indicate that the multiscale approach is very effective in exploring the statespace of the problem and providing feasible solutions of acceptable quality, while at the same it offers a significant acceleration. 1 Introduction The Hopfield neural network model [7, 8] and closely related models such as the Boltzmann Machine [3, 1] have proved effective in dealing with hard optimization problems and yield nearoptimal solutions with polynomial time complexity [6, 20]. The basic ...
Scalability of a neural network for the knightâs tour problem
 Neurocomputing
, 1996
"... The e ectiveness and e ciency of a Hop eldstyle neural network recently proposed by Takefuji and Lee for the knight's tour problem on an n n board are compared and contrasted with standard algorithmic techniques using a combination of experimental and theoretical analysis. Experiments indicate that ..."
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Cited by 2 (1 self)
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The e ectiveness and e ciency of a Hop eldstyle neural network recently proposed by Takefuji and Lee for the knight's tour problem on an n n board are compared and contrasted with standard algorithmic techniques using a combination of experimental and theoretical analysis. Experiments indicate that the neural network has poor performance when implemented on a conventional computer, and it is further argued that it is unlikely to improve signi cantly when implemented in parallel. Keywords: Knight's tour problem, neural network, parallel algorithm, Hamiltonian cycle problem.
EnergyBased Computation with Symmetric Hopfield Nets
"... We propose a unifying approach to the analysis of computational aspects of symmetric Hopfield nets which is based on the concept of "energy source". Within this framework we present different results concerning the computational power of various Hopfield model classes. It is shown that polynomial ..."
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We propose a unifying approach to the analysis of computational aspects of symmetric Hopfield nets which is based on the concept of "energy source". Within this framework we present different results concerning the computational power of various Hopfield model classes. It is shown that polynomialtime computations by nondeterministic Turing machines can be reduced to the process of minimizing the energy in Hopfield nets (the MIN ENERGY problem). Furthermore, external and internal sources of energy are distinguished. The external sources include e.g. energizing inputs from socalled Hopfield languages, and also certain external oscillators that prove finite analog Hopfield nets to be computationally Turing universal. On the other hand, the internal source of energy can be implemented by a symmetric clock subnetwork producing an exponential number of oscillations which are used to energize the simulation of convergent asymmetric networks by Hopfield nets. This shows that infinite families of polynomialsize Hopfield nets compute the complexity class PSPACE/poly. A special attention is paid to generalizing these results for analog states and continuous time to point out alternative sources of efficient computation. 1
General Purpose Computation with Neural Networks: A Survey of Complexity Theoretic Results
, 2003
"... We survey and summarize the existing literature on the computational aspects of neural network models, by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics. The criteria of classi cation include e.g. the architecture of the network (fee ..."
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We survey and summarize the existing literature on the computational aspects of neural network models, by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics. The criteria of classi cation include e.g. the architecture of the network (feedforward vs. recurrent), time model (discrete vs. continuous), state type (binary vs. analog), weight constraints (symmetric vs. asymmetric), network size ( nite nets vs. in  nite families), computation type (deterministic vs. probabilistic), etc.