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Symmetry definitions for constraint satisfaction problems. Constraints 11(2–3):115–137
, 2006
"... Abstract We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an opera ..."
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Cited by 61 (7 self)
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Abstract We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the kary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.
Solving the Kirkman's Schoolgirl Problem in a Few Seconds
 In Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming
, 2002
"... The Social Golfer Problem has been extensively used in recent years by the constraint community as an example of highly symmetric problem. It is an excellent problem for benchmarking symmetry breaking mechanisms such as SBDS or SBDD and for demonstrating the importance of the choice of the right mod ..."
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Cited by 30 (0 self)
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The Social Golfer Problem has been extensively used in recent years by the constraint community as an example of highly symmetric problem. It is an excellent problem for benchmarking symmetry breaking mechanisms such as SBDS or SBDD and for demonstrating the importance of the choice of the right model for one problem. We address in this paper a specific instance of the Golfer Problem well known as the Kirkman's Schoolgirl Problem and list a collection of techniques and tricks to find efficiently all its unique solutions. In particular, we propose SBDD+, an generic improvement over SBDD which allows a deep pruning when a symmetry is detected during the search. Our implementation of the presented techniques allows us to improve previous published results by an order of magnitude for CPU time as well as number of backtracks, and to compute the seven unique solutions of the Kirkman's problem in a few seconds.
Efficient Detection of Network Motifs
 IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS
, 2006
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Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. I. Permutation Groups and Coherent (Cellular) Algebras.
, 1997
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Graphical RuleBased Representation of SignalTransduction Networks
, 2005
"... The process by which a cell senses and responds to its environment, as in signal transduction, is often mediated by a network of proteinprotein interactions, in which proteins combine to form complexes and undergo posttranslational modifications, which regulate their enzymatic and binding activiti ..."
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Cited by 15 (7 self)
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The process by which a cell senses and responds to its environment, as in signal transduction, is often mediated by a network of proteinprotein interactions, in which proteins combine to form complexes and undergo posttranslational modifications, which regulate their enzymatic and binding activities. A typical signaling protein contains multiple sites of protein interaction and modification and may contain catalytic domains. As a result, interactions of signaling proteins have the potential to generate a combinatorially large number of complexes and modified states, and representing signaltransduction networks can be challenging. Representation, in the form of a diagram or model, usually involves a tradeoff between comprehensibility and precision: comprehensible representations tend to be ambiguous or incomplete, whereas precise representations, such as a long list of chemical species and reactions in a network, tend to be incomprehensible. Here, we develop conventions for representing signaltransduction networks that are both comprehensible and precise. Labeled nodes represent components of proteins and their states, and edges represent bonds between components. Binding and enzymatic reactions are described by reaction rules, in which left graphs define the properties of reactants and right graphs define the products that result from transformations of reactants. The reaction rules can be evaluated to derive a mathematical model.
Rulebased modeling of biochemical networks
 Complexity
, 2005
"... We present a method for generating a biochemical reaction network from a description of the interactions of components of biomolecules. The interactions are specified in the form of reaction rules, each of which defines a class of reaction associated with a type of interaction. Reactants within a cl ..."
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Cited by 10 (7 self)
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We present a method for generating a biochemical reaction network from a description of the interactions of components of biomolecules. The interactions are specified in the form of reaction rules, each of which defines a class of reaction associated with a type of interaction. Reactants within a class have shared properties, which are specified in the rule defining the class. A rule also provides a rate law, which governs each reaction in a class, and a template for transforming reactants into products. A set of reaction rules can be applied to a seed set of chemical species and, subsequently, any new species that are found as products of reactions to generate a list of reactions and a list of the chemical species that participate in these reactions, i.e., a reaction network, which can be translated into a mathematical model. © 2005 Wiley Periodicals, Inc. Complexity 10: 22–41, 2005 Key Words: local rules; automatic model generation; networks; signal transduction; combinatorial complexity; systems biology The cell is a complex adaptive system whose emergent behavior we understand only poorly. One reason for our lack of understanding is the complexity of cellular decision making, which is often mediated by a system of interacting proteins. Systems of interacting proteins are particularly prominent in signal transduction [1], 1 the focus Correspondence to: William S. Hlavacek,
Negative association in uniform forests and connected graphs, Random Structures Algorithms 24
 MR MR2060630 (2004m:60014
, 2004
"... Abstract. We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a unifor ..."
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Cited by 6 (0 self)
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Abstract. We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is verified numerically for all graphs G having eight or fewer vertices, or having nine vertices and no more than eighteen edges, using a certain computer algorithm which is summarised in this paper. Negative association is known already to be valid for a uniform spanning tree. The three cases of uniform forest, uniform spanning tree, and uniform connected subgraph are special cases of a more general conjecture arising from the randomcluster model of statistical mechanics. 1. Three random subgraphs Throughout this paper, G = (V, E) denotes a finite labelled graph with vertex set V and edge set E. An edge e with endpoints x, y is written e = 〈x, y〉. For reasons discussed at the end of Section 1, we shall assume that G has neither loops nor multiple
Formulation symmetries in circle packing
"... The performance of BranchandBound algorithms is severely impaired by the presence of symmetric optima in a given problem. We describe a method for the automatic detection of formulation symmetries in MINLP instances. A software implementation of this method is used to conjecture the group structur ..."
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Cited by 6 (1 self)
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The performance of BranchandBound algorithms is severely impaired by the presence of symmetric optima in a given problem. We describe a method for the automatic detection of formulation symmetries in MINLP instances. A software implementation of this method is used to conjecture the group structure of the problem symmetries of packing equal circles in a square. We provide a proof of the conjecture and compare the performance of spatial BranchandBound on the original problem with the performance on a reformulation that cuts away symmetric optima. Keywords: MINLP, spatial BranchandBound, Global Optimization, group, reformulation.
Reformulations in mathematical programming: Automatic symmetry detection and exploitation
 Mathematical Programming
"... symmetrydetection andexploitation ..."
Experimental Studies of the Universal Chemical Key (UCK) Algorithm on the NCI
 Database of Chemical Compounds, Proceedings of the 2003 IEEE Computer Society Bioinformatics Conference (CSB 2003), IEEE Computer Society, Los Alamitos
"... We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. T ..."
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Cited by 4 (3 self)
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We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. The algorithm was tested on 236,917 compounds obtained from the National Cancer Institute (NCI) database of chemical compounds. In this paper we present the algorithm, some examples and the experimental results on the NCI database. On the NCI database, the UCK algorithm provided distinct unique keys for chemicals with different molecular structures. 1.