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Many reasoning and optimization problems exhibit symmetries. Previous work has shown how special purpose algorithms can make use of these symmetries to simplify reasoning. We present a general scheme whereby symmetries are exploited by adding \symmetry-breaking" predicates to the theory. Our approach can be used on any propo-sitional satissability problem,… (More)

We announce an algebraic approach to the problem of assigning <italic>canonical forms</italic> to graphs. We compute canonical forms and the associated canonical labelings (or renumberings) in polynomial time for graphs of bounded valence, in moderately exponential, exp(n<supscrpt>½ + &ogr;(1)</supscrpt>),time for general graphs, in subexponential,… (More)

Suppose we are given a set of generators for a group G of permutations of a colored set A. The color automorphism problem for G involves finding generators for the subgroup of G which stabilizes the color classes. Testing isomorphism of graphs of valence ≤ t is polynomial-time reducible to the color automorphism problem for groups with small simple… (More)

The Constraint Satisfaction Problem is a type of combinatorial search problem of much interest in Artiicial Intelligence and Operations Research. The simplest algorithm for solving such a problem is chronological backtracking, but this method suuers from a malady known as \thrashing," in which essentially the same subproblems end up being solved repeatedly.… (More)

We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case 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… (More)

We conjecture that every nite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1). We show that it suuces to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for nite simple groups, such a short… (More)

A permutation group on n letters may always be represented by a small set of generators, even though its size may be exponential in n. We show that it is practical to use such a representation since many problems such as membership testing, equality testing, and inclusion testing are decidable in polynomial time. In addition, we demonstrate that the normal… (More)

We address the graph isomorphism problem and related fundamental complexity problems of computational group theory. The main results are these: A1. A polynomial time algorithm to test simplicity and find composition factors of a given permutation group (COMP). A2. A polynomial time algorithm to find elements of given prime order p in a permutation group of… (More)