L. Babai annd E. Luks. Canonical Labeling of Graphs. In 15th ACM Symposium on Theory of Computing, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....http: www.google.com hyperlinks Figure 3: A webpage as a semi structure. 3.1.1 The Labeling Issue Graphs present the issue of uniquely identifying and referencing nodes . For indistinguishable nodes, this is summarized in graph theory under the term canonical labeling [2] [3] [19] 21] and no solution has been provided with a high enough degree of generality. Nevertheless, content speci c watermarking techniques (for node content watermarking) as well as the new technique of keyed content hashing (see below) provides a resilient enough labeling scheme, suited for ....

L. Babai and E. Luks. Canonical labeling of graphs. In Fifteenth Annual ACM Symposium on Theory of Computing, pages 171-183. ACM, 1983.

....[5, p. 1461] No general worst case polynomial time algorithms are known for this problem, but it is commonly believed not to be NP complete (unless, of course, P=NP) 30] Polynomial time algorithms are available in many special cases [5, p. 1511] in particular for graphs of bounded degrees [33, 3]. Observe that many practical applications entail graphs of bounded degree because the objects involved (logic gates in VLSI chips, facts stored in knowledge bases, etc. are interconnected sparsely. In contrast, Boolean Satisfiability instances of bounded degree, e.g. 3 SAT, are known to be ....

L. Babai and E. M. Luks, "Canonical Labeling of Graphs", Symp. on the Theory of Computing (STOC) `83, pp. 171-183.

....as a University of Oregon doctoral student under the direction of Professor Eugene M. Luks. This paper represents on going work with Professor Luks, and the final version will appear elsewhere. Takunari Miyazaki lutions possible. Group theoretic methods led to the first significant result (cf. [3]; see also [12] Theorem 1.1 (Babai Klingsberg Luks) For vertex colored graphs of bounded color classes (i.e. color multiplicities) canonical forms can be computed in polynomial time. With considerably deeper use of group theory, Babai and Luks went on to show that canonical forms of ....

....For vertex colored graphs of bounded color classes (i.e. color multiplicities) canonical forms can be computed in polynomial time. With considerably deeper use of group theory, Babai and Luks went on to show that canonical forms of graphs of bounded degree can 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 ....

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L. BABAI and E. M. LUKS, Canonical labeling of graphs, Proceedings of the Fifteenth Annual ACM Symposium on the Theory of Computing, Boston, Apr. 25--27,

....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 ....

Laszlo Babai and Eugene M. Luks. Canonical labeling of graphs. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 171--183, Boston, Massachusetts, 25--27 April 1983.

....(smallest) string in an orbit of a given string is an FP NP complete problem under polynomial time many one reductions. This remains true even if the group acting on the string is an abelian 2 group. Our result is an improvement since the previous work only shows that the problem is NP hard [1]. In fact, the NP hardness result of Babai and Luks [1] shows that the problem of finding the lexicographically smallest (greatest) string in an orbit is at least as hard as finding the size of the largest independent set (clique) in a graph. When this is combined with the result of Krentel [10] ....

....an FP NP complete problem under polynomial time many one reductions. This remains true even if the group acting on the string is an abelian 2 group. Our result is an improvement since the previous work only shows that the problem is NP hard [1] In fact, the NP hardness result of Babai and Luks [1] shows that the problem of finding the lexicographically smallest (greatest) string in an orbit is at least as hard as finding the size of the largest independent set (clique) in a graph. When this is combined with the result of Krentel [10] published some years later, the constructive string ....

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L. Babai and E. M. Luks. Canonical labeling of graphs. In Fifteenth Annual ACM Symposium on Theory of Computing, pages 171--183. ACM, 1983.

....of graphs of bounded genus or bounded valence. Then there is a logic that e#ectively captures PTIME on C. Proof. It follows from results of Filotti and Mayer [22] and independently Miller [59] that there is a PTIME canonization function for isomorphism on graphs of bounded genus. Babai and Luks [4] proved that there is a PTIME canonization function for isomorphism on graphs of bounded valence. # 8.4. Definable canonization revisited. The disadvantage of the results in the previous subsection is that the logics we obtain are quite unnatural, even though they satisfy Gurevich s ....

L. Babai and E. M. Luks, Canonical labeling of graphs, Proceedings of the 15th ACM symposium on theory of computing, 1983, pp. 171--183.

....been intensively studied, in part because its many applications, an in part because it is one of the few problems in NP that has resisted all attempts to be classified as NP complete, or within P. The best existing upper bound for the problem given by Luks and Zemlyachenko is exp p cn log n (cf [7]) but there is no evidence of this bound being optimal, and for many restricted graph classes polynomial time algorithms are known. This is for example the case of planar graphs [14] graphs of bounded degree [23] or graphs with bounded eigenvalue multiplicity [6] In some cases, like trees [22, ....

L. Babai and E. Luks. Canonical labeling of graphs. In Proc. 15th ACM Symp on Theory of Computing pp. 171--183, 1983.

....input 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] Recently, Miltersen and Vinodchandran [MV99] managed to relax the hardness assumption in the last line of Table 1. They showed that AM collapses to NP if there exists a Boolean function in NE coNE with exponential nondeterministic circuit complexity (instead of SAT oracle circuit ....

L. Babai and E. Luks. Canonical labeling of graphs. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 171-183. ACM, 1983.

....of graphs of bounded genus or bounded valence. Then there is a logic that effectively captures PTIME on C. Proof. It follows from results of Filotti and Mayer [22] and independently Miller [59] that there is a PTIME canonization function for isomorphism on graphs of bounded genus. Babai and Luks [4] proved that there is a PTIME canonization function for isomorphism on graphs of bounded valence. 2 8 Recall the definition on page 35, right before Subsection 8.1. 38 MARTIN GROHE 8.4. Definable Canonization revisited. The disadvantage of the results in the previous subsection is that the ....

L. Babai and E.M. Luks. Canonical labeling of graphs. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 171--183, 1983.

....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 ....

Laszlo Babai and Eugene M. Luks. Canonical labeling of graphs. In Proc. 15th Annual ACM Symposium on Theory of Computing, pages 171--183, 1983.

....greatest marking symmetric to M . To classify the problem, we employ the problem CLIQUE SIZE asking the size of the largest clique in an undirected graph. 4 COMPLEXITY OF SUB PROBLEMS 9 Lemma 4.10 CLIQUE SIZE p m LEX GREATEST MARKING . Proof. We use a construction resembling the one by Babai and Luks [1983, Section 3.1] Given a non labelled undirected graph G = hV; Ei, construct the net N and marking MG for G as in the proof of Lemma 4.6. Now, assume an arbitrary total order V on the set V of vertices. De ne UL(v) f p v 0 ;v 00 j v 0 ; v 00 V vg (the set of places ....

....representative. Consequently, computing such cosets is a function problem at least as hard as GRAPH AUTOMORPHISMS. A similar concept is used in the graph automorphism tool NAUTY tool by McKay [1990] which computes the automorphism group and the canonical form of a graph at the same time. See also [Babai and Luks 1983] for a string canonization algorithm. 5 MARKING STABILIZERS 12 6 SYMMETRIC COVERABILITY We say that a marking M covers a marking M 0 if M 0 M . In order to build a coverability graph [Finkel 1990] of a net, we extend markings to be functions of form M : P (N [ f g) where is a symbol ....

BABAI, L. AND LUKS, E. M. 1983. Canonical labeling of graphs. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing. ACM, 171183.

....and its associated labeling the canonical labeling of X. A number of studies on computing canonical forms have shown considerable success. In particular, bounds on certain parameters of a graph make polynomial time solutions possible. Grouptheoretic methods led to the first significant result (cf. [3]; see also [12] Theorem 1.1 (Babai Klingsberg Luks) For vertex colored graphs of bounded color classes (i.e. color multiplicities) canonical forms can be computed in polynomial time. With considerably deeper use of group theory, Babai and Luks went on to show that canonical forms of graphs ....

....as the author s Directed Research Project as a University of Oregon doctoral student under the direction of Professor Eugene M. Luks. This paper represents on going work with Professor Luks, and the final version will appear elsewhere. University of Oregon, Eugene, 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 ....

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L. Babai and E. M. Luks, Canonical labeling of graphs, Proceedings of the 15th Annual ACM Symposium on the Theory of Computing, Boston, Apr. 25--27, 1983, ACM, New York, 1983, pp. 171--183.

....of graphs of bounded genus or bounded valence. Then there is a logic that effectively captures PTIME on C. Proof. It follows from results of Filotti and Mayer [22] and independently Miller [59] that there is a PTIME canonization function for isomorphism on graphs of bounded genus. Babai and Luks [4] proved that there is a PTIME canonization function for isomorphism on graphs of bounded valence. 2 8 Recall the definition on page 35, right before Subsection 8.1. 38 MARTIN GROHE 8.4. Definable Canonization revisited. The disadvantage of the results in the previous subsection is that the ....

L. Babai and E.M. Luks. Canonical labeling of graphs. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 171--183, 1983.

....by many researchers, including Faradzev, Zemlyachenko, Babai, and Mathon. With k = 1, this method gives a linear time graph isomorphism algorithm that works for almost all graphs [10] Furthermore, the fastest known general graph isomorphism algorithms make use of this method with k = O( p n) [11]. It had been conjectured that this method would provide a polynomial time graph isomorphism test at least for graphs of bounded valence. Valence is a synonym for degree. Our result disposes of such conjectures. Up until now, most lower bounds in this area were proved using random graphs. This ....

....in an exp( p n(log n) c ) isomorphism test. Subsequently Luks [33] proved, using group theory to greater depth, that isomorphism for graphs of bounded valence is in polynomial time. Finally the canonical labeling problem for graphs of bounded valence has been solved in polynomial time by [11] and [18] independently. Individualization followed by naive refinement has also been the tool used by Babai to handle strongly regular graphs [4] and primitive coherent configurations [6] He used individualization of k = 2 p n log n vertices. Strongly regular graphs and more generally, ....

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L'aszl'o Babai and Eugene M. Luks, "Canonical Labeling of Graphs," 15th ACM Symposium on Theory of Computing (1983), 171-183.

....input 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 ....

L. Babai and E. Luks. Canonical labeling of graphs. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 171--183. ACM, 1983.

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L. Babai and E.M. Luks, Canonical labeling of graphs, in: Proc. of 15th ACM Symp. on Theory of Computing (1983) pp. 171--183.

....are pleased to acknowledge partial support via NSF grant CCR9820945 The central theme of this paper is the eciency of constructing a suitable lex leader S. One discouraging obstacle is that, in general, it is unlikely that a succinct lex leader formula can be found. Indeed, Babai and Luks [1] show that the problem of testing lex leadership of a search space point is NPhard. Furthermore, this hardness persists even for abelian groups if one is not careful about the ordering of variables. Thus unless NP = P , there is no polynomial time algorithm that computes a lex leader formula ....

.... This exponential size is due to a combinatorial bottleneck which we formulate in terms of anti chains in a lattice and we tie in to a class of combinatorial objects, Sperner spaces, a generalization of Sperner families in extremal set theory [5, 14] On the positive side, however, results of [1] also demonstrated that, for a reasonable class of groups , there is a polynomial test for our desired lex leastness, though of course this may require some re ordering of the variables. That, in itself, guarantees a polynomial time constructible symmetry breaker since the algorithm can be ....

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L. Babai and E. M. Luks. Canonical labeling of graphs. In Proc. 15th ACM Symp. on Theory of Computing, pages 171-183, 1983.

....a linear system can be expressed as solvability of a dual linear system. As a result, we write (in Section 7.4) a lex leader formula LL (G) of size O(n ) for such groups (n = j) An additional obstacle arises in extending LL (G) to general abelian groups. Via a sharpening of a result of [1] and [4] we show that even if the orbits have size as large as 3, testing lex leastness of points (i.e. strings) is coNP complete (Section 6.2) Hence, it is quite unlikely that the property can be captured in a polynomial size formula. However, it seems that the problem is sensitive to the ....

....of one member of each equivalence class. 6 Theorem 5.3 The problem of testing whether a 0=1 string X is the lex leader in its G orbit is coNPcomplete. This is the case even if G is abelian with orbits of size 3. We remark that a slightly weaker result with orbits of size 4 can be deduced from [1] (Proposition 3.1) A result of this form was also noted in [4] Theorem 3.2) but the groups were nonabelian and the orbits unbounded. In that case the groups were explicitly constructed as the automorphisms of speci ed theories. It is possible, though less convenient in this case, to show how ....

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L. Babai and E. M. Luks. Canonical labeling of graphs. In Proc. 15th ACM Symp. on Theory of Computing, pages 171-183, 1983.

....suffice) Despite the succinctness of such representations, a substantial polynomial time machinery has developed for computing with permutation groups. A major stimulus for 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 ....

.... 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 generators. ....

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L. Babai and E.M. Luks, Canonical labeling of graphs, Proc. 15th ACM STOC, 1983, 171-183.

....tractable lex leader predicate since lex leader verification is NP hard. However, we can also directly construct theories where the symmetry tree cannot be pruned to polynomial size even though the lex leader problem is in polynomial time (the algorithm uses the string canonization procedure of [ Babai and Luks, 1983 ] applicable because the group turns out to be abelian) The existence of the polynomial time algorithm, in turn, guarantees we can find some symmetry breaking predicate in polynomial time even though SB(T ) is useless for this purpose. Details will appear in a future paper. 6 Approximation ....

L'aszl'o Babai and Eugene M. Luks. Canonical labeling of graphs. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 171--183, Boston, Massachusetts, 25--27 April 1983.

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L. Babai annd E. Luks. Canonical Labeling of Graphs. In 15th ACM Symposium on Theory of Computing, 1983.

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L. Babai annd E. Luks. Canonical Labeling of Graphs. In 15th ACM Symposium on Theory of Computing, 1983.

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L. Babai and E. M. Luks, "Canonical labeling of graphs," in Proc. Symp. Theory Comput., 1983, pp. 171--183.

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L. Babai and E. M. Luks. Canonical labeling of graphs. In Proc. 15th ACM Symposium on Theory of Computing, Association for Computing Machinery, New York NY, 1983, pp. 171--183.

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L. Babai and E. M. Luks. Canonical labeling of graphs. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 171--183. ACM, New York, 1983.

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