Babai, L.: Moderately exponential bounds for graph isomorphism. In: Proc. International Symposium on Fundamentals of Computing Theory 81. Volume 117 of Lecture Notes in Computer Science., Springer-Verlag, Berlin Heidelberg New York (1981) 34-50  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Theoretical Computer Science. Besides its many applications, the main source of interest in GI has been the evidence that this problem is probably neither in P nor NP complete. Other sources of interest include the sophistication of the tools developed to attack the problem (for example [3, 14]) and the connections between GI and structural complexity (see [10] Understandably, many GI restrictions have been considered. For example, P upper bounds are known in the cases of planar graphs [7] or graphs of bounded valence [14] In some cases, like trees [12, 4] or graphs with colored ....

Babai, L.: Moderately exponential bounds for graph isomorphism. In: Proc. International Symposium on Fundamentals of Computing Theory 81. Volume 117 of Lecture Notes in Computer Science., Springer-Verlag, Berlin Heidelberg New York (1981) 34-50

....in Theoretical Computer Science. Besides its many applications, the main source of interest in GI has been the evidence that this problem is probably neither in P nor NP complete. Other sources of interest include the sophistication of the tools developed to attack the problem (for example [3, 4]) and the connections between GI and structural complexity (see [5] Understandably, many GI restrictions have been considered. For example, P upper bounds are known in the cases of planar graphs [6] or graphs of bounded valence [4] In some cases, like trees [7, 8] or graphs with colored ....

L. Babai, Moderately exponential bounds for graph isomorphism, in: Proc. International Symposium on Fundamentals of Computing Theory 81, Vol. 117 of Lecture Notes in Computer Science, Springer-Verlag, Berlin Heidelberg New York, 1981, pp. 34-50.

.... be computed in polynomial time [3] and it was also shown independently by Furer, Schnyder, and Specker [8] On the other hand, the fastest known algorithm to compute canonical forms for general graphs runs in O(exp(n 1=2 o(1) time for n vertex graphs (Luks, Zemlyachenko, cf. 3] see also [1], 23] Older techniques were often based on a type of brute force backtrack search. In general, the naive brute force method searches all possible vertex labelings of an n vertex input graph, yielding O(n ) time. This naive approach can be improved by classifying the vertices of an input graph ....

....algorithm to exponential time, and one whose colors are ordered randomly. Our experimental results indicate the same exponential lower bound (obtained in x6.2, Proposition 6.2) 2. Preliminaries We begin with definitions and describe fundamentals of the naive vertex classification method (see [1], 2] 3] Throughout, all graphs we consider are finite and simple (i.e. undirected without loops and multiple edges) unless noted, and all colored graphs we consider are vertex colored graphs defined as follows. 2.1. Vertex coloring Let X = V; E) be a graph, where jV j = n. A coloring of ....

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L. BABAI, Moderately exponential bound for graph isomorphism, Fundamentals of Computation Theory, Proceedings of the

....were (and are still) concerned. Like during an epidemic more and more people were seized with this challenge, reports on some progress attracted a new crowd of researchers. Eventually, the situation was characterised as the graph isomorphism disease in [ReaC77] Gat79] There and in [ZemKT82] [Bab81], Hof82] ButL85] one can find discussions of many facets of the problem and detailed classifications of different approaches to its solution. Discussions of the problem through the eyes of chemists are found for example in [BonMB85] Gra86] LiuK91] For practically comparing graphs rigorous ....

Babai L.: Moderately exponential bound for graph isomorphism. Lecture Notes in Comput. Sci. 117, Springer, 1981, 34-50.

.... OR 1 2 TAKUNARI MIYAZAKI time [3] and it was also shown independently by Furer, Schnyder, and Specker [8] On the other hand, the fastest known algorithm to compute canonical forms for general graphs runs in O(exp(n 1=2 o(1) time for n vertex graphs (Luks, Zemlyachenko, cf. 3] see also [1], 23] Older techniques were often based on a type of brute force backtrack search. In general, the naive brute force method searches all possible vertex labelings of an n vertex input graph, yielding O(n ) time. This naive approach can be improved by classifying the vertices of an input graph ....

....algorithm to exponential time, and one whose colors are ordered randomly. Our experimental results indicate the same exponential lower bound (obtained in x6.2, Proposition 6.2) 2. Preliminaries We begin with definitions and describe fundamentals of the naive vertex classification method (see [1], 2] 3] Throughout, all graphs we consider are finite and simple (i.e. undirected without loops and multiple edges) unless noted, and all colored graphs we consider are vertex colored graphs defined as follows. 2.1. Vertex coloring. Let X = V; E) be a graph, where jV j = n. A coloring of ....

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L. Babai, Moderately exponential bound for graph isomorphism, Fundamentals of Computation Theory, Proceedings of the 1981 International FCT-Conference, Szeged, Aug. 24--28, 1981 (F. G'ecseg, ed.), Lecture Notes in Comput. Sci., vol. 117, Springer, New York, 1981, pp. 34--50.

.... the base cases 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 ....

.... 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 procedures identifying elements or subgroups are constructive i.e. computed via straight line ....

L. Babai, Moderately exponential bound for graph isomorphism, Proc. Conf. FCT 1981, Szeged, Springer Lect. Notes in Comp. Sci. 117 (1981), 34--50.

....maximum degree 1 of G, and check whether H is isomorphic to each k connected component. Zemlyachenko, Korneenko, and Tyshkevich showed the algorithm solving the Graph Isomorphism problem in O(e n 10c ) time, where c is a positive constant less than 1 [15] The result is reviewed by Babai in [3]. Using their algorithm, we can solve the problem in O i nt1 (k) k e k 10c j time. When 1 3, e k 10c is less than 1 k . When 1 3, each graph is either a path or a cycle. Then the isomorphism of the graphs is easy to check in polynomial time. Thus, in each case, we can check the ....

L. Babai. Moderately Exponential Bound for Graph Isomorphism. In Fundamentals of Computer Theory, pages 34--50. Lecture Notes in Computer Science Vol. 117, Springer-Verlag, 1981.

....theorems about finite groups such as the fact that groups of odd order are solvable. In this paper we are motivated by the question, Are there algorithms that run in time n O(logn) or better whose running time bounds can be obtained purely combinatorially In a paper on graph isomorphism, Babai[1] describes a technique of Zemlyachenko which attempts to break the symmetry between vertices of a graph by picking a subset, S, of distinguished vertices and partitioning the rest of the vertices based on degree to the distinguished vertices and recursively on degrees to sets in the partition. ....

....a set S then we have an O(n jSj ) algorithm for the isomorphism question since we simply try all possible sets of size upto jSj in both graphs and try to establish an isomorphism 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) ....

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L. Babai, Moderately Exponential Bound for Graph Isomorphism, FCT 3 , 1981.

....to be in co NP. It is well known that if P 6= NP then there exists problems which are of intermediate status, and it has been hypothesized that GI may be one of these problems. One early result on the complexity of GI showed that there is a moderately exponential algorithm for solving the problem [3]. Moderately exponential means that on a problem of size n, the running time t obeys the relation: n k t a n , where k; a 1 are arbitrary real numbers. Typically this means the exponent in the running time is a function of n which grows slower than n. For example, the one upper bound ....

L. Babai. Moderately exponential bound for graph isomorphism. Proceedings of the Fundamentals of Computing Science, Lecture Notes in Computing Science 117, pages 34--50, 1981.

....testing whether two given graphs are isomorphic has withstood all attempts for a solution up to date. The worst case running time of all known algorithms is of exponential order, and just for certain special types of graphs, polynomial time algorithms have been devised (for further reference see [1, 22, 25]) Although the possibility that Graph Isomorphism (GI) is NP complete has been discussed [14] strong evidence against this possibility has been provided [29, 4, 8, 16, 18, 33] In the first place it was proved by Mathon [29] that the decision and counting versions of the problem are ....

L. Babai, Moderately exponential bound for graph isomorphism, in Proc. Fundamentals of Computation Theory Conference, Lecture Notes in Computer Science #117 (1981), 34--50.

....In the following, we will give a brief overview of graph and subgraph isomorphism algorithms. There are two basic approaches that past research has taken towards the problem of graph isomorphism. The first approach is based on group theoretic concepts and the study of permutation groups. In [Bab81], it was shown that there exists a moderately exponential bound for the general graph isomorphism problem. Furthermore, by imposing certain restrictions on the properties of the graphs, it was possible to derive algorithms that have a polynomially bounded complexity. For example, Luks and ....

L. Babai. Moderately exponential bound for graph isomorphism. In F. Gecseg, editor, Lecture Notes in Computer Science: Fundamentals of Computation Theory, pages 34--50. Springer Verlag, 1981.

....2 dim W L produces necessarily a primitive coherent configuration. For tournaments, the isomorphism problems of primitive and arbitrary coherent configurations are polynomial time equivalent [11] The general graph isomorphism problem has been attacked by Zemlyachenko. The method is described in [5] and [41] By individualization of O( p n) vertices and canonical edgeswitching, he has been able to reduce the valence to O( p n) Combining this with the method of Luks [33] Zemlyachenko obtained the first interesting upper bound for general graph isomorphism [41] cf. 5] His bound is ....

.... is described in [5] and [41] By individualization of O( p n) vertices and canonical edgeswitching, he has been able to reduce the valence to O( p n) Combining this with the method of Luks [33] Zemlyachenko obtained the first interesting upper bound for general graph isomorphism [41] cf. [5]) His bound is exp(n 1 Gammac ) for some positive constant c. This has subsequently been improved by Babai and Luks [11] to exp(n 1=2 o(1) Instead of measuring the reduction in the valence, one could ask about the effect of these methods on the color class size. Babai [7] has investigated ....

L'aszl'o Babai, "Moderately Exponential Bound for Graph Isomorphism," Proc. Conf. on Fundamentals of Computation Theory, Lecture Notes in Computer Science, Springer, 1981, 34-50.

....color and either both adjacent, or both not adjacent to the other chosen pair. Thus Player II can always preserve the partial isomorphism. 2 1. 8 Vertex Refinement Corresponds to C 2 It turns out that the expressive power of C 2 is characterized by the well known method of vertex refinement (see [2, 16]) Let G = hV; E; C 1 ; C r i 1. Describing Graphs: a First Order Approach to Graph Canonization 17 be a colored graph in which every vertex statisfies exactly one color relation. Let f : V f1 : ng be given by f(v) i iff v 2 C i . We then define f 0 , the refinement of f as ....

Laszlo Babai, "Moderately Exponential Bound for Graph Isomorphism, " Proc. Conf. on Fundamentals of Computation Theory, Szeged, August 1981.

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L. Babai. Moderately exponential bound for graph isomorphism. In F. Gcseg, editor, Fundamentals of Computation Theory, FCT'81, volume 117 of Lecture Notes in Computer Science, pages 34-50. Springer-Verlag, 1981. 9

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L. Babai. Moderately exponential bounds for graph isomorphism. In Fundamentals of Computation Theory 81, Lecture Notes in Computer Science #117, Springer-Verlag, pp. 34--50, 1981.

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