Completeness results for graph isomorphism

Jenner, Birgit; Köbler, Johannes; McKenzie, Pierre; Torán, Jacobo

doi:10.1016/s0022-0000(03)00042-4

Cited by 48 publications

(44 citation statements)

References 23 publications

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“…The strongest known hardness result [40] says that GI is hard for DET, which is a subclass of NC 2 . The complexity status of GI is determined precisely only if the problem is restricted to trees: For trees GI is LOGSPACE-complete [28,25].…”

Section: The Graph Isomorphism Problemmentioning

confidence: 99%

Testing Graph Isomorphism in Parallel by Playing a Game

Grohe

Verbitsky

2006

Automata, Languages and Programming

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Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in TC 1 ⊆ NC 2 . If no counting quantifiers are needed, then Graph Isomorphism for C is even in AC 1 . The definability conditions can be checked by designing a winning strategy for suitable Ehrenfeucht-Fraïssé games with a logarithmic number of rounds. The parallel isomorphism algorithm this approach yields is a simple combinatorial algorithm known as the Weisfeiler-Lehman (WL) algorithm.Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC 1 , answering an open question from [13]. Furthermore, we obtain an AC 1 algorithm for testing isomorphism of rotation systems (combinatorial specifications of graph embeddings). The AC 1 upper bound was known before, but the fact that this bound can be achieved by the simple WL algorithm is new. Combined with other known results, it also yields a new AC 1 isomorphism algorithm for planar graphs.

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Section: The Graph Isomorphism Problemmentioning

confidence: 99%

Testing Graph Isomorphism in Parallel by Playing a Game

Grohe

Verbitsky

2006

Automata, Languages and Programming

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“…Proof: If A is many-one logarithmic space reducible to GI via a function f , then the set Bit f belongs to L. The result follows from Theorem 3.3 since it is known that every set in L is strongly many-one reducible to PGI [6,10].…”

Section: Corollary 34mentioning

confidence: 87%

“…It is known that every set in NC 1 , L, NL and in several other complexity classes is strong many-one AC 0 reducible to PGI [6,10].…”

Section: Pgi = {((G H) (I J))| G H If and Only If I J}}mentioning

confidence: 99%

Reductions to Graph Isomorphism

Torán

2007

FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science

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“…We explain our motivation for studying the space complexity of k-tree isomorphism. On the one hand, we have Lindell's result [Lin92,JKMT03] that tree canonization is complete for deterministic logspace, 1 which tightly characterizes the complexity of both isomorphism and canonization of trees. What about partial k-tree isomorphism?…”

Section: Introductionmentioning

confidence: 99%