Hoffmann, C.M., Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science, Vol. 136, Springer-Verlag, New York, NY (1982).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of G not mapping v to w, is Turing equivalent to the graph isomorphism problem. In contrast to this, she showed that deciding whether a graph has a fix point free automorphism is NP complete. For further information about the graph isomorphism problem, we refer to the books by Hoffmann [14], and by Kobler, Schoning Toran [18] as well as to the surveys by Read Corneil [28] Babai [2] and Fortin [10] Overview. In Sect. 2, we show that the (combinatorial) polytope isomorphism problem is graph isomorphism complete (Theorem 2) This remains true for simple (every vertex figure is ....

Hoffmann, C.: Group-theoretic Algorithms and Graph Isomorphism (Lecture Notes in Computer Science 136) Berlin-Heidelberg-New York: Springer 1982

....of G not mapping v to w, is Turing equivalent to the graph isomorphism problem. In contrast to this, she showed that deciding whether a graph has a fix point free automorphism is NP complete. For further information about the graph isomorphism problem, we refer to the books by Hoffmann [14], and by Kobler, Schoning Toran [18] as well as to the surveys by Read Corneil [28] Babai [2] and Fortin [10] Overview. In Sect. 2, we show that the (combinatorial) polytope isomorphism problem is graph isomorphism complete (Theorem 2) This remains true for simple (every vertex figure is ....

Hoffmann, C.: Group-theoretic Algorithms and Graph Isomorphism (Lecture Notes in Computer Science 136) Berlin-Heidelberg-New York: Springer 1982

.... The automorphisms of a given graph Gamma can be studied with respect to several points of view: one can study the general structure of every automorphism and classify them with respect to this structure (see [Hal73] for example) one can study the structure of the group of automorphisms (see [Hof82] in the case of finite graphs) and more generally the structure of any group acting on Gamma (see [BS83] for trees, DD90] for graphs) At last, one might study the structure of the quotient Hn Gamma of Gamma by a group H acting on Gamma (which seems to be a less classical subject for graphs ....

C.M. Hoffmann. Group-theoretic algorithms and graph-isomorphism. LNCS 136, 1982.

....We now digress for a bit to consider the complexity of the distinguishing question. First we observe that D(G) 1 if and only if G is a rigid graph, i.e. one whose automorphism group is trivial. The complexity of deciding if a given graph has a non trivial automorphism has not been settled [9, 11]. It is known to be Turing equivalent to Unique Graph Isomorphism, and is a candidate for a problem whose difficulty lies between being in P and being NP Gamma complete. Hence determining if D(G) 1 may be difficult. Let us fix the particular question to be: Given a graph G and an integer k, is ....

Christoph Hoffmann, Group Theoretic Algorithms and Graph Isomorphism, Springer Verlag, New York, 1982 (Lecture Notes in Computer Science 136).

....We now digress for a bit to consider the complexity of the distinguishing question. First we observe that D(G) 1 if and only if G is a rigid graph, i.e. one whose automorphism group is trivial. The complexity of deciding if a given graph has a non trivial automorphism has not been settled [9, 11]. It is known to be Turing equivalent to Unique Graph Isomorphism, and is a candidate for a problem whose diculty lies between being in P and being NP complete. Hence determining if D(G) 1 may be dicult. Let us x the particular question to be: Given a graph G and an integer k, is D(G) k For ....

Christoph Homann, Group Theoretic Algorithms and Graph Isomorphism, Springer Verlag, New York, 1982 (Lecture Notes in Computer Science 136).

....whether a tuple y0 belongs to a group represented by a set of generators. Consider the tower of subgroups G i , 0 i n where G i is the subgroup of J obtained fixing the first i components to 1. A right coset representation of this tower can be efficiently constructed using algorithm 7 in [12] 1 . Now we will present a polynomial time algorithm that, given a right coset representation for the tower of subgroups, decides whether y0 2 G0 , proving the polynomial time evaluability of CG operations. Let a1 ; ar be the right coset representatives of G0 in G1 . Let T = fa i : 1 ....

M.C. Hoffmann. Group-Theoretic Algorithms and Graph Isomorphism, volume 136 of Lecture Notes in Computer Science. SpringerVerlag, 1982.

....as needed for the argument at hand. Theorem 32 The general subgroup problem for a finite group G is polynomially solvable. Proof. The result follows immediately from a known algorithm that finds generators for a group, obtain by Babai [6] Furst et al. 17] see also Theorem II.12 in Hoffmann [27]. The main observation is that given a group H with known generators and a chain of subgroups H = H 0 H 1 Delta Delta Delta H r = f1g, one can obtain distinct representatives from each coset of each H i in H i Gamma1 as follows. Select two elements x, x 0 among the generators of H 0 ....

....there is only one generator in each coset of each H i in H i Gamma1 , and carry out the process for products xy of two current generators as well. The fact that only products of pairs are needed to obtain representatives for all cosets of each H i in H i Gamma1 requires proof, see Theorem II.8 in [27]. In our application, we have n elements that must be assigned values in G, so a solution is an element of H = G n . Each relation in an instance defines a subgroup or more generally a coset a i J i in H , for some subgroup J i of H , with 1 i s if there are s relations in an instance. Let H ....

[Article contains additional citation context not shown here]

C. M. Hoffmann, "Group-Theoretic Algorithms and Graph Isomorphism," Lecture Notes in Comp. Sci. 136 (1982), Springer-Verlag.

....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 and ....

Hoffmann C.M.: Group-theoretic algorithms and graph isomorphism. Lecture Notes in Comput. Sci. 136, Springer, Berlin, 1982.

....hidden subgroup states are distinguishable in a small number of oracle calls. A number of problems are reducible to hidden subgroup problems including discrete logarithm, graph isomorphism, code equivalence, and various equivalent problems thought to be strictly harder than graph isomorphism [6]. As an example of this last category we mention restricted graph automorphism, where given a graph Gamma on n vertices and a subgroup J of S n given by generators one should find a set of generators for the subgroup Aut( Gamma) J . It is well known [7] that when G is Abelian one can ....

Hoffmann, Christoph M. Group-theoretic algorithms and graph isomorphism. Lecture Notes in Computer Science, Vol. 136. SpringerVerlag, New York, 1982.

....can be done in FP AM and particularly, for group definable languages over the group family is SYM, counting the number of solutions can be done in FP NP with parallel queries. As an intermediate result, we prove upper bounds on the following generalization of a problem posed by Hoffman [Ho82]: Given a polynomial time membership test for a subgroup H of a group G (presented by a set of generators) to compute a set of generators for H. An interesting aspect of this problem is that it is a kind of converse of the usual problem of membership testing in a group presented by a generator ....

C. Hoffmann. Group-Theoretic Algorithms and Graph Isomorphism. Lecture Notes in Computer Science, bf 136 (Springer-Verlag 1982).

....n , compute a generator set for A B. Actually, Group Intersection is easily reducible to Group Intersection Generators. We also show that Group Intersection Generators has a nonadaptive NC checker. Our proof techniques use fairly simple ideas from algorithmic permutation group theory (see e.g. [19, 10, 13, 7]) The plan of the paper is as follows. In Section 2 we give definitions, and in Section 3 we show that the search problem for GINT can be solved in parallel with nonadaptive queries. In Section 4 we describe the above mentioned 2 round interactive protocol for GINT . Finally, in Section 5 we ....

....: g k ) x) g j (x) if x 2 X j . As a useful example, notice that for G S n and X [n] the stabilizer group GX can be expressed as the intersection of G with the direct sum SX Phi S ( n] GammaX) We next define the wreath product of a group G S n with the group S 2 . 2 Definition 7 [13] For G S n the wreath product (G) of G with S 2 , is a permutation group on [n] Theta [2] The elements of (G) are written as (g 1 ; g 2 ; for g 1 ; g 2 2 G and 2 S 2 , where the permutation defined by (g 1 ; g 2 ; is as below: i. If = id then (g 1 ; g 2 ; hi; ji = hg j (i) ji, 8i ....

[Article contains additional citation context not shown here]

C. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science #136, Springer, 1982.

....We now digress for a bit to consider the complexity of the distinguishing question. First we observe that D(G) 1 if and only if G is a rigid graph, i.e. one whose automorphism group is trivial. The complexity of deciding if a given graph has a non trivial automorphism has not been settled [9, 11]. It is known to be Turing equivalent to Unique Graph Isomorphism, and is a candidate for a problem whose di#culty lies between being in P and being NP complete. Hence determining if D(G) 1 may be di#cult. Let us fix the particular question to be: Given a graph G andanintegerk,isD(G) k For k ....

Christoph Ho#mann, Group Theoretic Algorithms and Graph Isomorphism, Springer Verlag, New York, 1982 (Lecture Notes in Computer Science 136).

.... The automorphisms of a given graph Gamma can be studied with respect to several points of view: one can study the general structure of every automorphism and classify them with respect to this structure (see [Hal73] for example) one can study the structure of the group of automorphisms (see [Hof82] in the case of finite graphs) and more generally the structure of any group acting on Gamma (see [BS83] for trees, DD90] for graphs) At last, one might study the structure of the quotient Hn Gamma of Gamma by a group H acting on Gamma (which seems to be a less classical subject for graphs ....

C.M. Hoffmann. Group-theoretic algorithms and graph-isomorphism. LNCS 136, 1982.

....We also give a lower bound for BI: we show that the problem of deciding if a graph has a unique optimal clique which is not known to be in the Boolean Hierarchy many one reduces to it. The Boolean Isomorphism problem shares many similarities with the Graph Isomorphism problem, GI (see [Hof82] and [KST93] for a comprehensive study on Graph Isomorphism) Many of the results for GI carry over to BI with similar proofs, although with some crucial differences. We also consider some more general transformations of the input. An important such generalization is that of Boolean congruence, ....

C. Hoffmann. Group-theoretic algorithms and graph isomorphism. Springer Verlag, Lecture Notes in Computer Sience 136, 1982.

....When K = 1, P3(i) is contained in [FHL] the analogue for quotient groups is immediate since h(H=K) G=K i = hH G i=K. P3(ii) is an easy consequence of the observation that H is subnormal in G iff H is subnormal in hH G i. When K = 1, P4(i) is an easy application of results in [FHL] see [Ho], or [CFL] where it is directly reduced to 3.2) the general case is an immediate consequence. In view of P3(ii) P4(ii) follows at once. For K = 1, P4(iii) is in [Lu1, x4.2] The general case is solved herein, see x7. All parts of P4 should be compared with the general problem INTERSECTION (see ....

C.M. Hoffmann, Group Theoretic Algorithms and Graph Isomorphism, Lect. Notes in Comp. Sci. 136, Springer 1982.

....show in this paper, it turns out that we need some nontrivial permutation group theory in order to design a nonadaptive NC checker for Group Intersection. In particular, in Section 4. we introduce a novel notion of wreath product of permutation groups generalizing the well known wreath product [12]. Using wreath products we are also able to design a nonadaptive NC checker also for Group Intersection Generators, improving the result in [3] 1 In [1] actually an efficient constant query sequential checker is designed, which also turns out to be implementable in NC. We now summarize ....

....S n be a permutation group, and for some X [n] let GX denote the setwise stabilizer of X in G. Then GX can be expressed as the intersection of G with the direct sum SX Phi S ( n] GammaX) We next define the wreath product of any group G S n with the permutation group S 2 . 2 Definition 7 [12] Let G S n be some permutation group. The wreath product of G and S 2 , which we denote by (G) is a permutation group that acts on the set [n] Theta [2] The elements of (G) are written as (g 1 ; g 2 ; for g 1 ; g 2 2 G and 2 S 2 , where the permutation defined by (g 1 ; g 2 ; ....

[Article contains additional citation context not shown here]

C. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science #136, Springer, 1982.

....and g is defined analogously. Next we present two problems from group theory which are generator equivalent. For this we need some notions from elementary group theory. For a more detailed introduction to group theory, we refer to [Hum96, Cam95] Our reductions are based on the ones given in [Hof82]. S n denotes the symmetric group of f1; ng, i.e. the set of all permutations over f1; ng. For a set A S n , we denote by hAi the smallest subgroup of S n which contains A. A is called a generator set for hAi. It is known that every subgroup of S n can be generated by a set of at ....

C.M. Hoffmann. Group-Theoretic Algorithms and Graph Isomorphism. Lecture Notes in Computer Science #136, Springer, 1982.

....which is in fact an equality problem, is trivially solvable in polynomial time. The graph ismorphism problem is therefore in NP, but is not NP complete, unless the polynomial hierarchy collapses [BHZ87] see also [Sch89] comprehensive studies on the graph ismorphism problem can be found in [Hof82, KST93]) The equivalence of two deterministic finite automatas (DFA) can be decided in polynomial time. It is not hard to see that the isomorphism problem for DFA s, where one can permute the states of a DFA, is still solvable in polynomial time. For a subclass of the branching programs, the ....

C. Hoffmann. Group-theoretic algorithms and graph isomorphism. Lecture Notes in Computer Sience 136 , Springer Verlag, 1982.

....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 ....

....GI is the best known example of a problem with this property, it is not an isolated case. There are many other graph and group theoretic problems related to Graph Isomorphism that lie between P and NP complete and whose exact 2 Kobler, Schoning Tor an complexity is not known either (see [22, 23, 28]) These problems can be divided into different classes depending on whether they seem easier or harder than GI: problems which are reducible to GI (like Graph Automorphism) problems Turing equivalent to GI (so called isomorphism complete problems) problems to which GI is Turing reducible (like ....

[Article contains additional citation context not shown here]

C. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science #136, Springer, 1982.

No context found.

Hoffmann, C.M., Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science, Vol. 136, Springer-Verlag, New York, NY (1982).

No context found.

C. M. Hoffmann. Group-Theoretic Algorithms and Graph Isomorphism. Springer-Verlag, New York NY, 1982.

No context found.

Hoffmann, C.M., Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science, Vol. 136, Springer-Verlag, New York, NY (1982).

No context found.

C. Hoffmann. Group-theoretic algorithms and graph isomorphism. Lecture Notes in Computer Science 136. Springer Verlag, 1981.

No context found.

C. Hoffmann, Group-theoretic algorithms and graph isomorphism, Springer LNCS vol. 136 (1979).

No context found.

C.M. Hoffmann. Group-Theoretic Algorithms and Graph Isomorphism, Vol. 136 of the series, Lecture Notes in Computer Science, ed. G. Goos and J. Hartmanis, Springer-Verlag, Berlin (1982).

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC