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Computation in Permutation Groups: Counting and Randomly Sampling Orbits
"... Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the or ..."
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Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the orbits (exactly), (2) approximately count the orbits, and (3) choose an orbit uniformly at random. The goal is to quantify the computational diculty of the problems. In particular, we would like to know for which input sets I the problems are tractable.
Polynomialdelay enumeration of monotonic graph classes
 Journal of Machine Learning Research
"... Algorithms that list graphs such that no two listed graphs are isomorphic, are important building blocks of systems for mining and learning in graphs. Algorithms are already known that solve this problem efficiently for many classes of graphs of restricted topology, such as trees. In this article we ..."
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Algorithms that list graphs such that no two listed graphs are isomorphic, are important building blocks of systems for mining and learning in graphs. Algorithms are already known that solve this problem efficiently for many classes of graphs of restricted topology, such as trees. In this article we introduce the concept of a dense augmentation schema, and introduce an algorithm that can be used to enumerate any class of graphs with polynomial delay, as long as the class of graphs can be described using a monotonic predicate operating on a dense augmentation schema. In practice this means that this is the first enumeration algorithm that can be applied theoretically efficiently in any frequent subgraph mining algorithm, and that this algorithm generalizes to situations beyond the standard frequent subgraph mining setting.
Randomly Sampling Unlabelled Structures
, 1999
"... Informally, an \unlabelled combinatorial structure" is an object such as an unlabelled graph (in which the vertices are indistinguishable) or a structural isomer in chemistry (in which dierent atoms of the same type are indistinguishable). Computational experiments such as those described in ..."
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Informally, an \unlabelled combinatorial structure" is an object such as an unlabelled graph (in which the vertices are indistinguishable) or a structural isomer in chemistry (in which dierent atoms of the same type are indistinguishable). Computational experiments such as those described in this volume often rely on random sampling to generate inputs for the experiments. This paper surveys work on the problem of eciently sampling unlabelled combinatorial structures from a uniform distribution. 1 Introduction Most of the experimental work described in this volume involves rst randomly sampling combinatorial structures and second using the randomlychosen structures as inputs to computational experiments. In order for the experiments to be valid, the distribution from which the combinatorial structures are drawn must be precisely specied. In order for the experiments to be computationally feasible, the randomsampling algorithms must be ecient. This survey is devoted to the ...
General graph refinement with polynomial delay
 In Proceedings of the Workshop on Machine Learning and Graphs (MLG’07
, 2007
"... Of many graph mining algorithms an essential component is its procedure for enumerating graphs such that no two enumerated graphs are isomorphic. All frequent subgraph miners require such a component [14, 5, 1, 6], but also other ..."
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Of many graph mining algorithms an essential component is its procedure for enumerating graphs such that no two enumerated graphs are isomorphic. All frequent subgraph miners require such a component [14, 5, 1, 6], but also other
Algorithms for Group Actions: Homomorphism Principle and Orderly Generation Applied to Graphs
 OF DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1996
"... The generation of discrete structures up to isomorphism is interesting as well for theoretical as for practical purposes. Mathematicians want to look at and analyse structures and for example chemical industry uses mathematical generators of isomers for structure elucidation. The example chosen in t ..."
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The generation of discrete structures up to isomorphism is interesting as well for theoretical as for practical purposes. Mathematicians want to look at and analyse structures and for example chemical industry uses mathematical generators of isomers for structure elucidation. The example chosen in this paper for explaining general generation methods is a relatively far reaching and fast graph generator which should serve as a basis for the next more powerful version of MOLGEN, our generator of chemical isomers.
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
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Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
Efficient generation of representative systems
, 2005
"... Abstract. Often, it is beneficial to have lists of all representative elements of a set of combinatorial configurations. For instance, one may wish to have a list of all nonisomorphic graphs with n vertices, or a list of nonisomorphic subspaces of a vector space. In this paper, we present Read’s a ..."
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Abstract. Often, it is beneficial to have lists of all representative elements of a set of combinatorial configurations. For instance, one may wish to have a list of all nonisomorphic graphs with n vertices, or a list of nonisomorphic subspaces of a vector space. In this paper, we present Read’s algorithm to generate such lists. This algorithm is considerably more efficient than the naive approach, and can be modified to produce a wide variety of combinatorial configurations. Consider the problem of generating all nonisomorphic graphs or digraphs on n vertices. Or that of generating all partitions of an nset or free trees with n vertices. Pólya’s enumeration theory gives an elegant method to determine the number of such configurations, but fails to provide us with the configurations themselves. In this paper, we study Read’s orderly algorithm to generate all unique combinatorial configurations of a certain type [Rea78, CR79b, CR79a]. This algorithm improves on the “classical ” method to generate such lists by introducing techniques to eliminate the need to search through existing configurations when considering a newly generated configuration. Given a set, W, with a finite number of equivalence classes, we can use this orderly
Lehrstuhl II für Mathematik (Informatik)
, 2003
"... The generation of molecular graphs by computer programs has undergone some changes. The development is reported with focus on various mathematical methods that are created and employed in this process. No attempt has been made to explicitely state and prove the theorems but this overview contains hi ..."
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The generation of molecular graphs by computer programs has undergone some changes. The development is reported with focus on various mathematical methods that are created and employed in this process. No attempt has been made to explicitely state and prove the theorems but this overview contains hints to the relevant literature. In particular, a new generator MOLGEN 4.0 is described that aims at highest flexibility in using constraints during the generation process and, thus, meeting the needs of practical applications.