L. Babai, W.M. Kantor,and E.M. Luks, Computational complexity and the classification of finite simple groups, Proc. 24th IEEE Symp. Found. Comp. Sci. (1983), 162-171.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....derandomization of AM (that is, a proof of the statement AM=NP) would immediately give polynomial size membership proofs for positive instances of Graph Nonisomorphism. In contrast, the lengths of the shortest proofs known, without any assumptions, are exponential in the sizes of the graphs [BL83, BKL83] In [AK97] Arvind and Kobler showed that the construction of [NW94] can be extended to the nondeterministic setting to get pseudorandom generators which can be used to completely derandomize AM. As in the case of [NW94] they needed an average case hardness assumption in order to construct the ....

Laszlo Babai, William M. Kantor, and Eugene M. Luks. Computational complexity and the classification of finite simple groups. In 24th Annual Symposium on Foundations of Computer Science, pages 162--171, Los Alamitos, Ca., USA, November 1983. IEEE

....derandomization of AM (that is, a proof of the statement AM=NP) would immediately give polynomial size membership proofs for positive instances of Graph Nonisomorphism. In contrast, the lengths of the shortest proofs known, without any assumptions, are exponential in the sizes of the graphs [BL83, BKL83] In [AK97] Arvind and Kobler showed that the construction of [NW94] can be extended to the nondeterministic setting to get pseudorandom generators which can be used to completely derandomize AM. As in the case of [NW94] they needed an average case hardness assumption in order to construct the ....

Laszlo Babai, William M. Kantor, and Eugene M. Luks. Computational complexity and the classification of finite simple groups. In Proc. 24th Annual IEEE Symposium on Foundations of Computer Science, pages 162--171, 1983.

....diameter of permutation groups. Jerrum [15] showed that the problem of computing the shortest sequence of generators for arbitrary permutation groups is PSPACE complete, even when there are only two generators. The general problem has seen considerable attention see the survey of Babai, et al. [2]. The outline of this paper is as follows. In Section 2 we present our notation for describing sequences of reversal operations. In Section 3, we present our notion of equivalent transformations, which is the primary tool we need to understand sorting with fixed length reversals. Armed with these ....

L. Babai, W. M. Kantor, and E. M. Luks. Computational complexity and the classification of finite simple groups. In Proc. 24th IEEE Symposium on Foundations of Computer Science (FOCS), pages 162--171, 1983.

....based on a 1 1 correspondence between vertices in the two sets. For the case of arbitrary graphs the technique in [1] yields a bound of exp(n 2=3 o(1) for the isomorphism of n node graphs. Subsequent papers have improved upon this time bound using other techniques. See for example [2]. In this paper we show that a very simple algorithm for tournament isomorphism along the lines of [1] has an n O( p n) time bound. We also define the notion of an anchor in a tournament to be a set of nodes which distinguishes every other pair of nodes at distance 1. This is a major ....

L. Babai, W.M. Kantor, and E.M. Luks, Computational Complexity and the Classification of Finite Simple Groups, Proc. FOCS 1983, 162--171.

....a simple normal subgroup T . Further, either jT j = r and K is a subgroup of the automorphism group of T Theta T , or K is a subgroup of the automorphism group of T . Following the Classification of Finite Simple groups, much is known about the permutation representations of finite simple groups [2]. In particular, either T is one of polynomially many alternating or classical groups (each of which can be described by a string of length O(log n) or jKj is polynomially bounded. In the first case jKj=jT j is polynomially bounded, so in either case O(log n) bits suffice to describe the order ....

....implies SAT 2 P, is not known. While no polynomial time algorithms for GI or #GA have been discovered, algorithms with subexponential running time do exist. For example, there exists an algorithm with time complexity exp(O( p n log n) which computes the automorphism group and its generators [2] (this is harder than solving GI and #GA) This algorithm combines the techniques of several authors including Babai, Luks, and Zemlyachenko. The bound on the running time is the same as the upper bound on the enumerability of #GA that we achieved in Section 6. This striking observation brings up ....

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L. Babai, W. M. Kantor, and E. M. Luks. Computational complexity and the classification of finite simple groups. In Proceedings of the 24th IEEE Symposium Foundations of Computer Science, pp. 162--171, 1983.

....diameter of permutation groups. Jerrum [16] showed that the problem of computing the shortest set of generators for arbitrary permutation groups is PSPACE complete, even when there are only two generators. The general problem has seen considerable attention see the survey of Babai, et al. [3]. The outline of this paper is as follows. In section 2 we present our notation for describing sequences of reversal operations. In section 3, we present our notion of equivalent transformations, which is the primary tool we need to understand sorting with fixed length reversals. Armed with these ....

L. Babai, W. M. Kantor, and E. M. Luks. Computational complexity and the classification of finite simple groups. In Proc. 24th IEEE Symposium on Foundations of Computer Science (FOCS), pages 162--171, 1983.

....lengths unless the polynomial time hierarchy collapses. Without any unproven hypothesis, the smallest proofs known for the nonisomorphism of two graphs on n vertices are of size 2 O( p n log n) namely the transcripts of the deterministic graph isomorphism algorithm by Babai et al. BL83, BKL83] Our simulations of AM are a special case of an all purpose derandomization tool which applies to any randomized process for which we can efficiently check the successfulness of a given random bit sequence. We formally define the notion of a success predicate in Section 4. If we can decide the ....

L. Babai, W. Kantor, and E. Luks. Computational complexity and the classification of finite simple groups. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 162--171. IEEE, 1983.

....a simple and general class of canonization algorithms. We hope that variants of these algorithms will be powerful enough to provide simple canonical forms for all graphs; and do so without resorting to the the high powered group theory needed in the present, best graph isomorphism algorithms [27, 3]. 1.2 Descriptive Complexity In this section we discuss an alternate view of complexity in which the complexity of the descriptions of problems is measured. This approach has provided new insights and techniques to help us understand the standard complexity notions: time, memory space, parallel ....

L. Babai, W.M. Kantor, E.M. Luks, "Computational Complexity and the Classification of Finite Simple Groups," 24th IEEE FOCS Symp., (1983), 162-171.

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L. Babai, W.M. Kantor,and E.M. Luks, Computational complexity and the classification of finite simple groups, Proc. 24th IEEE Symp. Found. Comp. Sci. (1983), 162-171.

....that easily exploited orbits and, in the transitive case, used the primitive action on a block system to break the group into a small number of cosets of the intransitive stabilizer of the blocks. With care, and some additional tricks, the method can be shown to run in time O(n cd= log d ) [4]. This method is routinely used in our normalizer algorithm as well. But we further develop an analog of this divide and conquer paradigm for matrix group computation, since the normalizer problem even for permutation groups naturally leads to instances of finding subspace stabilizers in certain ....

L. BABAI, W. M. KANTOR, and E. M. LUKS, Computational complexity and the classification of finite simple groups, 24th Annual Symposium on Foundations of Computer Science, Tucson, Nov. 7--9,

....this activity was the application to the graph isomorphism problem (ISO) for early work ( Ba1] FHL] Lu1] Mi1] Mi2] BL] used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], Lu2] Ne] KT] Ka1] Ka2] Ka3] BLS1] making available constructive versions of standard theoretical tools. One essential ingredient has, to a great extent, been lacking. The facility to deal with quotient groups (equivalently, homomorphic images of groups) is a central methodology of ....

.... in certain divide and conquer procedures; see, e.g. Lu1] There are fairly elementary procedures for testing membership in Gamma d (see [Lu1, x4] For our purposes, it is essential only that d be fixed; the specific value of d would play a role in more precise timing arguments [Ba2] BL] [BKL]) The class Gamma d arose originally in the context of testing graph isomorphism ( Lu1] Ba2] Mi1] Mi2] BL] FSS] 3. Algorithmic preliminaries Unless indicated otherwise, subgroups of Sym(n) Sym( Omega Gamma are input via generators. Output of groups is always via generators. All ....

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L. Babai, W.M. Kantor and E.M. Luks, Computational complexity and the classification of finite simple groups, Proc. 24th IEEE FOCS, 1983, 162--171.

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L. Babai, W. M. Kantor and E. M. Luks, Computational complexity and the classification of finite simple groups, pp. 162--171 in: Proc. IEEE Symp. on Foundations of Computer Science 1983.

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L.Babai, W.M.Kantor and E.M.Luks, Computational complexity and the classification of finite simple groups, 24th IEEE Symposium on Foundations of Comp.Sci., (1983), 162--171.

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L. Babai, W.M. Kantor, and E.M. Luks, "Computational Complexity and the Classification of Finite Simple Groups," 24th IEEE Symp. on Foundations of Computer Science (1983), 162-171.

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