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Algorithmic Data Science (Invited Talk)

Petra Mutzel, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science
The authors show that if the homomorphism vectors are restricted to trees, then the feature vectors of the homomorphism counts of two graphs are identical if and only if the Weisfeiler-Leman algorithm  ...  Basic Weisfeiler-Leman The Weisfeiler-Leman algorithm (WL) simultaneously colours the vertices of the two given graphs iteratively. In the beginning, all vertices get the same colour c.  ...

Lovász Meets Weisfeiler and Leman [article]

Holger Dell, Martin Grohe, Gaurav Rattan
2018 arXiv   pre-print
We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our  ...  we relate a beautiful theory by Lov\'asz with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman  ...  XX:4 Lovász Meets Weisfeiler and Leman Figure 1 Two fractionally non-isomorphic graphs with the same path homomorphism counts. Theorem 2.  ...

Isomorphism Testing for Graphs Excluding Small Minors

Martin Grohe, Daniel Wiebking, Daniel Neuen
2020 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
Weisfeiler-Leman Algorithm The Weisfeiler-Leman algorithm, originally introduced by Weisfeiler and Leman in its twodimensional form  , forms one of the most fundamental subroutines in the context  ...  Next, we define the 2-dimensional Weisfeiler-Leman algorithm.  ...  So let i ∈ [r] and consider the bipartite graph By the properties of the 2-dimensional Weisfeiler-Leman algorithm the graph B is biregular.  ...

Isomorphism Testing for Graphs Excluding Small Minors [article]

Martin Grohe, Daniel Neuen, Daniel Wiebking
2020 arXiv   pre-print
Weisfeiler-Leman Algorithm The Weisfeiler-Leman algorithm, originally introduced by Weisfeiler and Leman in its twodimensional form  , forms one of the most fundamental subroutines in the context  ...  Next, we define the 2-dimensional Weisfeiler-Leman algorithm.  ...  So let i ∈ [r] and consider the bipartite graph By the properties of the 2-dimensional Weisfeiler-Leman algorithm the graph B is biregular.  ...

Lovász Meets Weisfeiler and Leman

Holger Dell, Martin Grohe, Gaurav Rattan
2018
We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our  ...  paper, we relate a beautiful theory by Lovász with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman  ...  Since Lovász Meets Weisfeiler and Leman Now suppose HOM r (G) = HOM r (H) holds, and set v = HOM r (G).  ...

Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs [article]

Laura Mančinska, David E. Roberson
2019 arXiv   pre-print
Over 50 years ago, Lovász proved that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph [Acta Math. Hungar. 18 (1967), pp. 321–328].  ...  As there exist pairs of non-isomorphic graphs that are quantum isomorphic, this implies that homomorphism counts from planar graphs do not determine a graph up to isomorphism.  ...  Therefore if G ∼ = P H, then G and H are not distinguished by 2-dimensional Weisfeiler-Leman.  ...

Graph isomorphism: Physical resources, optimization models, and algebraic characterizations [article]

Laura Mančinska, David E. Roberson, Antonios Varvitsiotis
2020 arXiv   pre-print
As it turns out, the notion of DN N -isomorphism coincides with an equivalence relation on graphs introduced in 1968 by Weisfeiler and Leman  , known today as the 2-dimensional Weisfeiler-Leman method  ...  Otherwise, we say that the graphs are not distinguished by the Weisfeiler-Leman method, which is an equivalence relation on graphs.  ...

The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs [article]

Sandra Kiefer, Daniel Neuen
2021 arXiv   pre-print
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components.  ...  In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of the class of graphs of treewidth at most k.  ...  The Weisfeiler-Leman Algorithm Let χ, χ ′ : V k → C be colorings of the k-tuples of vertices of G, where C is some finite set of colors.  ...

Graph Isomorphism in Quasipolynomial Time [article]

László Babai
2016 arXiv   pre-print
Luks's barrier situation is characterized by a homomorphism ϕ that maps a given permutation group G onto S_k or A_k, the symmetric or alternating group of degree k, where k is not too small.  ...  Weisfeiler-Leman canonical refinement Classical WL refinement The classical Weisfeiler-Leman 5 (WL) refinement [WeL, We] takes as input a binary configuration and refines it to a coherent configuration  ...  Coherent configurations are the stable configurations of the classical Weisfeiler-Leman canonical refinement process [WeL, We] R ⊆ Ω × Ω let R − = {(y, x) | (x, y) ∈ R}.  ...

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed [article]

Laszlo Babai
2021 arXiv   pre-print
If X = (V, E) is a graph then the Weisfeiler-Leman color refinement process [WL68, We76] (see, e. g., [Ba16] ) efficiently constructs a CC X(X) = (V, c) such that Aut(X) = Aut(X(X)).  ...  The homomorphism ϕ ii is the identity on G i . The ϕ i,j are called the transition homomorphisms.  ...

Nonlocal Games and Quantum Permutation Groups [article]

Martino Lupini, Laura Mančinska, David E. Roberson
2018 arXiv   pre-print
Part of this work was completed during the Focused Research Meeting at Queen's University Belfast in October 2017, which was funded by the Heilbronn Institute for Mathematical Research.  ...  We remark that this coherent algebra can be computed in polynomial time via the (2-dimensional) Weisfeiler-Leman method  .  ...  It is also known that two graphs are equivalent if and only if they cannot be distinguished by the (2-dimensional) Weisfeiler-Leman algorithm.  ...

Between primitive and 2-transitive: Synchronization and its friends

João Araújo, Peter Cameron, Benjamin Steinberg
2017 EMS Surveys in Mathematical Sciences
Coherent configurations were introduced independently by Donald Higman [53, 54] in the USA and by Weisfeiler and Leman  in the former Soviet Union to describe the orbits on pairs of a permutation  ...  Now 3-sets in the same S(L) meet in 0 or 2 points. So any image of the Fano plane meets each S(L) in at most (and hence exactly) one set.  ...

On the automorphism groups of strongly regular graphs II

László Babai
2015 Journal of Algebra
The Weisfeiler-Leman refinement [61, 60] defines a refined coloring c by making c (x, y) = c (u, v) if and only if c(x, y) = c(u, v) and for all j, k < r we have p(x, y; i, j) = p(u, v; i, j).  ...  This map is a homomorphism, given that F is a functor. Proof of Theorem 22. Suppose (f, F ) is a functorial reduction from GI to the isomorphism problem for s. r. graphs.  ...

On the automorphism groups of strongly regular graphs I

László Babai
2014 Proceedings of the 5th conference on Innovations in theoretical computer science - ITCS '14
The Weisfeiler-Leman refinement [61, 60] defines a refined coloring c by making c (x, y) = c (u, v) if and only if c(x, y) = c(u, v) and for all j, k < r we have p(x, y; i, j) = p(u, v; i, j).  ...  This map is a homomorphism, given that F is a functor. Proof of Theorem 22. Suppose (f, F ) is a functorial reduction from GI to the isomorphism problem for s. r. graphs.  ...

СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ Special issue: Graphs and Groups, Spectra and Symmetries-G2S2 2016 ON GRAPHS AND GROUPS, SPECTRA AND SYMMETRIES HELD ON AUGUST 15-28, 2016, NOVOSIBIRSK, RUSSIA

E Konstantinova, D Krotov, A Mednykh, Rina Khomyakova, Elena Konstantinova, Denis Krotov, Alexey Medvedev, E Konstantinova, D Krotov, A Mednykh
2016 unpublished
The importance of this notion was explained by the fact that if the coherent configuration of a graph is separable, then the isomorphism of this graph to any other graph can be tested by the Weisfeiler-Leman  ...  They as well as the young students emphasized how glad they were that mathematicians from so many countries had come all the way to Siberia to meet their Russian colleagues.  ...
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