An optimal lower bound on the number of variables for graph identification

Cai, Jin-Yi; Fürer, Martin; Immerman, Neil

doi:10.1007/bf01305232

Cited by 467 publications

(703 citation statements)

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“…Color stabilization is now reached in r < 2n k rounds. The k-dim WL is polynomial-time for each constant k. In 1990, Cai, Fürer, and Immerman [12] proved a striking negative result: For any sublinear dimension k = o(n), the k-dim WL does not work correctly even on graphs of vertex degree 3. Nevertheless, later it was realized that a constant-dimensional WL is still applicable to particular classes of graphs, including planar graphs [19], graphs of bounded genus [20], and graphs of bounded treewidth [21].…”

Section: The Multidimensional Weisfeiler-lehman Algorithmmentioning

confidence: 99%

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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 Multidimensional Weisfeiler-lehman Algorithmmentioning

confidence: 99%

“…For the history of this approach to GI we refer the reader to [5,12,16,17]. We will abbreviate k-dimensional Weisfeiler-Lehman algorithm by k-dim WL.…”

Section: The Multidimensional Weisfeiler-lehman Algorithmmentioning

confidence: 99%

Testing Graph Isomorphism in Parallel by Playing a Game

Grohe

Verbitsky

2006

Automata, Languages and Programming

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“…At one time it was conjectured that FPC, the extension of inflationary fixed-point logic by counting terms, would suffice to express all polynomial-time properties, but this was refuted by Cai, Fürer and Immerman [CFI92]. Since then, a number of logics have been proposed whose expressive power is strictly greater than that of FPC but still contained within P. Among these are FPR, fixed-point logic with rank operators [DGHL09], andCPT(Card), choiceless polynomial time with counting [BGS99,BGS02].…”

Section: Introductionmentioning

confidence: 99%

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Anderson

Dawar

Holm

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC.This settles an open problem first posed by Blass, Gurevich and Shelah [BGS99], who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by showing that the ellipsoid method for solving linear programs can be implemented in FPC. This allows us to prove that linear programs can be optimised in FPC if the corresponding separation oracle problem can be defined in FPC. On the way to defining a suitable separation oracle for the maximum matching problem, we provide FPC formulas defining maximum flows and canonical minimum cuts in capacitated graphs.

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“…In the k-dimensional version, which we call k-WL in the following, we colour k-tuples of vertices instead of single vertices. For a detailed description of the algorithm and its history, we refer the reader to [8]. Using similar tricks as for colour refinement, one can implement k-WL to run in time O(n k log n) [17].…”

Section: Introductionmentioning

confidence: 99%

Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement

Berkholz

Bonsma

Grohe

2013

Lecture Notes in Computer Science

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An assignment of colours to the vertices of a graph is stable if any two vertices of the same colour have identically coloured neighbourhoods. The goal of colour refinement is to find a stable colouring that uses a minimum number of colours. This is a widely used subroutine for graph isomorphism testing algorithms, since any automorphism needs to be colour preserving. We give an O((m + n) log n) algorithm for finding a canonical version of such a stable colouring, on graphs with n vertices and m edges. We show that no faster algorithm is possible, under some modest assumptions about the type of algorithm, which captures all known colour refinement algorithms.

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An optimal lower bound on the number of variables for graph identification

Cited by 467 publications

References 30 publications

Testing Graph Isomorphism in Parallel by Playing a Game

Testing Graph Isomorphism in Parallel by Playing a Game

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement

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