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11
Pruning by Isomorphism in BranchandCut
, 2001
"... The paper presents a BranchandCut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non standard, cutting integer feasible solutions but leaving unchanged the optimal value of the probl ..."
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Cited by 30 (1 self)
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The paper presents a BranchandCut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non standard, cutting integer feasible solutions but leaving unchanged the optimal value of the problem. Pruning and cut generation are performed by backtracking procedures using a SchreierSims table for representing the group. Applications to the generation of covering designs and error correcting codes are presented.
Fast management of permutation groups I
, 1997
"... We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play ..."
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Cited by 21 (3 self)
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We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.
Combinatorial Tools for Computational Group Theory
 Groups and Computation, DIMACS Ser. Discrete Math. Theoret. Comput. Sci
, 1993
"... Abstract. A variety of elementary combinatorial techniques for permutation groups are reviewed. It is shown how to apply these techniques to yield faster and/or more spaceefficient algorithms for problems including group membership, normal closure, center, base change and Cayley graphs. Emphasis is ..."
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Cited by 21 (5 self)
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Abstract. A variety of elementary combinatorial techniques for permutation groups are reviewed. It is shown how to apply these techniques to yield faster and/or more spaceefficient algorithms for problems including group membership, normal closure, center, base change and Cayley graphs. Emphasis is placed on randomized techniques and new data structures. The paper includes both a survey of recent algorithms with which the authors have been associated, and some new algorithms in the same spirit that have not previously appeared in print. Many of the results include both complexity bounds and pseudocode, along with comments for faster software implementations.
New Methods for Using Cayley Graphs in Interconnection Networks
 DISCRETE APPLIED MATHEMATICS
, 1992
"... A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more timeefficient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayl ..."
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Cited by 17 (5 self)
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A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more timeefficient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayley graphs in a spaceefficient manner (log 2 (3) bits per node), so that copies could be cheaply stored at each node of an interconnection network. The second algorithm is especially useful for providing a compact encoding of an optimal routing table (for example, a 13 kilobyte optimal table for 64,000 nodes). The algorithm relies on using a compact encoding of group elements known from computational group theory. Generalizations of all of the above are presented for Schreier coset graphs.
Exploiting symmetry on parallel architectures
, 1995
"... This thesis describes techniques for the design of parallel programs that solvewellstructured problems with inherent symmetry. Part I demonstrates the reduction of such problems to generalized matrix multiplication by a groupequivariant matrix. Fast techniques for this multiplication are described ..."
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Cited by 10 (1 self)
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This thesis describes techniques for the design of parallel programs that solvewellstructured problems with inherent symmetry. Part I demonstrates the reduction of such problems to generalized matrix multiplication by a groupequivariant matrix. Fast techniques for this multiplication are described, including factorization, orbit decomposition, and Fourier transforms over nite groups. Our algorithms entail interaction between two symmetry groups: one arising at the software level from the problem's symmetry and the other arising at the hardware level from the processors' communication network. Part II illustrates the applicability of our symmetryexploitation techniques by presenting a series of case studies of the design and implementation of parallel programs. First, a parallel program that solves chess endgames by factorization of an associated dihedral groupequivariant matrix is described. This code runs faster than previous serial programs, and discovered a number of results. Second, parallel algorithms for Fourier transforms for nite groups are developed, and preliminary parallel implementations for group transforms of dihedral and of symmetric groups are described. Applications in learning, vision, pattern recognition, and statistics are proposed. Third, parallel implementations solving several computational science problems are described, including the direct nbody problem, convolutions arising from molecular biology, and some communication primitives such as broadcast and reduce. Some of our implementations ran orders of magnitude faster than previous techniques, and were used in the investigation of various physical phenomena.
Group membership for groups with primitive orbits, this volume
"... Abstract. This paper considers a permutation group G = 〈S 〉 of degree n with t orbits such that the action on each orbit is primitive. It presents a O(tn 2 log c (n)) time Monte Carlo group membership algorithm for some constant c. The algorithm is notable for its use of a new theorem showing how to ..."
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Cited by 1 (1 self)
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Abstract. This paper considers a permutation group G = 〈S 〉 of degree n with t orbits such that the action on each orbit is primitive. It presents a O(tn 2 log c (n)) time Monte Carlo group membership algorithm for some constant c. The algorithm is notable for its use of a new theorem showing how to find O(tlog 2 n) generators in O˜(Sn) time under a more general form of the above hypotheses. The algorithm relies on new combinatorial methods for computing with groups [CF92] and previous work of Babai, Luks and Seress [BLS88]. In addition, it makes extensive use of a structure theorem for primitive groups by Cameron [Cam81], which can be derived from results of Kantor [Kan79] and the classification of finite simple groups. 1.
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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Cited by 1 (1 self)
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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.
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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Cited by 1 (1 self)
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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.
Pruning by Isomorphism in BranchandCut
, 2001
"... Abstract The paper presents a BranchandCut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non standard, cutting integer feasible solutions but leaving unchanged the optimal value of ..."
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Abstract The paper presents a BranchandCut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non standard, cutting integer feasible solutions but leaving unchanged the optimal value of the problem. Pruning and cut generation are performed by backtracking procedures using a SchreierSims table for representing the group. Applications to the generation of covering designs and error correcting codes are presented. 1 Introduction Let \Pi n be the set of all permutations of the ground set In = f1; : : : ; ng. A permutation in \Pi n is represented by an nvector ss, with ss[i] being the image of i under ss. If v is an nvector and ss 2 \Pi n, let w = ss(v) denote the vector w obtained by permuting the coordinates of v according to ss, i.e. w[ss[i]] = v[i] for all i 2 In:
Let
"... this paper, we assume that an ILP together with its symmetry group G is given. WeshowhowtouseG in order to prune eciently isomorphic subproblems and to help the search by generating isomorphism cuts (cutting integer feasible solutions, but leaving the value of the optimal solution unchanged). This i ..."
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this paper, we assume that an ILP together with its symmetry group G is given. WeshowhowtouseG in order to prune eciently isomorphic subproblems and to help the search by generating isomorphism cuts (cutting integer feasible solutions, but leaving the value of the optimal solution unchanged). This isomorphism pruning is compatible with standard cut generation techniques (Gomory cuts, LiftandProject cuts, or specially designed cuts for the problem at hand). The price to pay for the pruning is that the branching variable can no longer be chosen arbitrarily. We also assume that the reader is familiar with the B&C procedure, as an excellentintroduction can be found in [23], [27], [28]