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Marc Hellmuth - One of the best experts on this subject based on the ideXlab platform.
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local algorithms for the Prime Factorization of strong product graphs
arXiv: Discrete Mathematics, 2017Co-Authors: Marc Hellmuth, Wilfried Imrich, Werner Klockl, Peter F StadlerAbstract:The practical application of graph Prime Factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant "approximate" Prime Factorization, is the development of local approaches that cover the graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear al- gorithm for the Prime Factorization of "locally unrefined" graphs with respect to the strong product. To this end we introduce the backbone B(G) for a given graph G and show that the neighborhoods of the backbone vertices provide enough information to determine the global Prime factors.
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a local Prime factor decomposition algorithm for strong product graphs
arXiv: Discrete Mathematics, 2017Co-Authors: Marc HellmuthAbstract:This work is concerned with the Prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived. Moreover, since most graphs are Prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known "classical" Prime Factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products.
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the cartesian product of hypergraphs
Journal of Graph Theory, 2012Co-Authors: Lydia Ostermeier, Marc Hellmuth, Peter F StadlerAbstract:We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique Prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and Žerovnik for the case of graphs and introduces the notion of diagonal-free grids as a replacement of the chord-free 4-cycles that play a crucial role in the case of graphs. This leads to a generalization of relation Δ on the arc set, whose convex hull is shown to coincide with the product relation of the Prime Factorization. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.
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a local Prime factor decomposition algorithm
Discrete Mathematics, 2011Co-Authors: Marc HellmuthAbstract:This work is concerned with the Prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived. Moreover, since most graphs are Prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known ''classical'' Prime Factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products.
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local Prime factor decompositionof approximate strong product graphs
2010Co-Authors: Marc HellmuthAbstract:In practice, we observe perturbed product structures, so-called approximate graph products, since structures derived from real-life data are notoriously incomplete and/or plagued by measurement errors. In fact, a very small perturbation, such as a deletion or insertion of a single edge, can destroy the product structure completely, modifying a product graph to a Prime graph. The practical application of the well-known Prime Factorization algorithms is therefore limited, since most graphs are Prime, although they can have a product-like structure. In order to deal with such inaccuracies, a mathematical framework is needed that allows us to deal with graphs that are only approximate products.
Peter F Stadler - One of the best experts on this subject based on the ideXlab platform.
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local algorithms for the Prime Factorization of strong product graphs
arXiv: Discrete Mathematics, 2017Co-Authors: Marc Hellmuth, Wilfried Imrich, Werner Klockl, Peter F StadlerAbstract:The practical application of graph Prime Factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant "approximate" Prime Factorization, is the development of local approaches that cover the graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear al- gorithm for the Prime Factorization of "locally unrefined" graphs with respect to the strong product. To this end we introduce the backbone B(G) for a given graph G and show that the neighborhoods of the backbone vertices provide enough information to determine the global Prime factors.
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the cartesian product of hypergraphs
Journal of Graph Theory, 2012Co-Authors: Lydia Ostermeier, Marc Hellmuth, Peter F StadlerAbstract:We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique Prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and Žerovnik for the case of graphs and introduces the notion of diagonal-free grids as a replacement of the chord-free 4-cycles that play a crucial role in the case of graphs. This leads to a generalization of relation Δ on the arc set, whose convex hull is shown to coincide with the product relation of the Prime Factorization. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.
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a Prime factor theorem for a generalized direct product
Discussiones Mathematicae Graph Theory, 2006Co-Authors: Wilfried Imrich, Peter F StadlerAbstract:We introduce the concept of neighborhood systems as a generalization of directed, re∞exive graphs and show that the Prime Factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.
Massimiliano Di Ventra - One of the best experts on this subject based on the ideXlab platform.
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polynomial time solution of Prime Factorization and np complete problems with digital memcomputing machines
Chaos, 2017Co-Authors: Fabio L Traversa, Massimiliano Di VentraAbstract:We introduce a class of digital machines, we name Digital Memcomputing Machines, (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space, and energy). An abstract DMM with this power must satisfy a set of compatible mathematical constraints underlying its practical realization. We prove this by making a connection with the dynamical systems theory. This leads us to a set of physical constraints for poly-resource resolvability. Once the mathematical requirements have been assessed, we propose a practical scheme to solve the above class of problems based on the novel concept of self-organizing logic gates and circuits (SOLCs). These are logic gates and circuits able to accept input signals from any terminal, without distinction between conventional input and output terminals. They can solve boolean problems by self-organizing into their solution. They can be fabricated either with circuit elements with memory (such as memristors) and/or standard MOS technology. Using tools of functional analysis, we prove mathematically the following constraints for the poly-resource resolvability: (i) SOLCs possess a global attractor; (ii) their only equilibrium points are the solutions of the problems to solve; (iii) the system converges exponentially fast to the solutions; (iv) the equilibrium convergence rate scales at most polynomially with input size. We finally provide arguments that periodic orbits and strange attractors cannot coexist with equilibria. As examples, we show how to solve the Prime Factorization and the search version of the NP-complete subset-sum problem. Since DMMs map integers into integers, they are robust against noise and hence scalable. We finally discuss the implications of the DMM realization through SOLCs to the NP = P question related to constraints of poly-resources resolvability.
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polynomial time solution of Prime Factorization and np hard problems with digital memcomputing machines
arXiv: Emerging Technologies, 2015Co-Authors: Fabio L Traversa, Massimiliano Di VentraAbstract:We introduce a class of digital machines we name Digital Memcomputing Machines (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space and energy). An abstract DMM with this power must satisfy a set of compatible mathematical constraints underlying its practical realization. We initially prove this by introducing the complexity classes for these machines. We then make a connection with dynamical systems theory. This leads to the set of physical constraints for poly-resource resolvability. Once the mathematical requirements have been assessed, we propose a practical scheme to solve the above class of problems based on the novel concept of self-organizing logic gates and circuits (SOLCs). These are logic gates and circuits able to accept input signals from any terminal, without distinction between conventional input and output terminals. They can solve boolean problems by self-organizing into their solution. They can be fabricated either with circuit elements with memory (such as memristors) and/or standard MOS technology. Using tools of functional analysis, we prove mathematically the following constraints for the poly-resource resolvability: i) SOLCs possess a global attractor; ii) their only equilibrium points are the solutions of the problems to solve; iii) the system converges exponentially fast to the solutions; iv) the equilibrium convergence rate scales at most polynomially with input size. We finally provide arguments that periodic orbits and strange attractors cannot coexist with equilibria. As examples we show how to solve the Prime Factorization and the NP-hard version of the subset-sum problem. Since DMMs map integers into integers they are robust against noise, and hence scalable. We finally discuss the implications of the DMM realization through SOLCs to the NP=P question related to...
Yusuke Isono - One of the best experts on this subject based on the ideXlab platform.
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unique Prime Factorization for infinite tensor product factors
arXiv: Operator Algebras, 2017Co-Authors: Yusuke IsonoAbstract:In this article, we investigate a unique Prime Factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse free product components. In the type III setting, this is the first classification result for infinite tensor product non-amenable factors. Our proof is based on Popa's intertwining techniques and a characterization of relative amenability on the continuous cores.
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unique Prime Factorization and bicentralizer problem for a class of type iii factors
Advances in Mathematics, 2017Co-Authors: Cyril Houdayer, Yusuke IsonoAbstract:Abstract We show that whenever m ≥ 1 and M 1 , … , M m are nonamenable factors in a large class of von Neumann algebras that we call C ( AO ) and which contains all free Araki–Woods factors, the tensor product factor M 1 ⊗ ‾ ⋯ ⊗ ‾ M m retains the integer m and each factor M i up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [33] , [25] and moreover provides new UPF results in the case when M 1 , … , M m are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type I I I 1 factors in the class C ( AO ) .
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unique Prime Factorization and bicentralizer problem for a class of type iii factors
arXiv: Operator Algebras, 2015Co-Authors: Cyril Houdayer, Yusuke IsonoAbstract:We show that whenever $m \geq 1$ and $M_1, \dots, M_m$ are nonamenable factors in a large class of von Neumann algebras that we call $\mathcal C_{(\text{AO})}$ and which contains all free Araki-Woods factors, the tensor product factor $M_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} M_m$ retains the integer $m$ and each factor $M_i$ up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [OP03, Is14] and moreover provides new UPF results in the case when $M_1, \dots, M_m$ are free Araki-Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type ${\rm III_1}$ factors in the class $\mathcal C_{(\text{AO})}$.
Daniel Drimbe - One of the best experts on this subject based on the ideXlab platform.
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Prime ii1 factors arising from actions of product groups
Journal of Functional Analysis, 2020Co-Authors: Daniel DrimbeAbstract:Abstract We prove that any II1 factor arising from a free ergodic probability measure preserving action Γ ↷ X of a product Γ = Γ 1 × … × Γ n of icc hyperbolic, free product or wreath product groups is Prime, provided Γ i ↷ X is ergodic, for any 1 ≤ i ≤ n . We also completely classify all the tensor product decompositions of a II1 factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique Prime Factorization result for such II1 factors. Finally, we obtain a unique Prime Factorization theorem for a large class of II1 factors which have property Gamma.
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Prime ii _1 factors arising from actions of product groups
arXiv: Operator Algebras, 2019Co-Authors: Daniel DrimbeAbstract:We prove that any II$_1$ factor arising from a free ergodic probability measure preserving action $\Gamma\curvearrowright X$ of a product $\Gamma=\Gamma_1\times\dots\times\Gamma_n$ of icc hyperbolic, free product or wreath product groups is Prime, provided $\Gamma_i\curvearrowright X$ is ergodic, for any $1\leq i\leq n.$ We also completely classify all the tensor product decompositions of a II$_1$ factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique Prime Factorization result for such II$_1$ factors. Finally, we obtain a unique Prime Factorization theorem for a large class of II$_1$ factors which have property Gamma.