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Documents authored by Köbler, Johannes

Document
DOI: 10.4230/LIPIcs.STACS.2024.6
On a Hierarchy of Spectral Invariants for Graphs

Authors: V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer’s question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.
Cite as

V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On a Hierarchy of Spectral Invariants for Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2024.6,
  author =	{Arvind, V. and Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg},
  title =	{{On a Hierarchy of Spectral Invariants for Graphs}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.6},
  URN =		{urn:nbn:de:0030-drops-197166},
  doi =		{10.4230/LIPIcs.STACS.2024.6},
  annote =	{Keywords: Graph Isomorphism, spectra of graphs, combinatorial refinement, strongly regular graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.43
Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Authors: Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.
Cite as

Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fuhlbruck_et_al:LIPIcs.STACS.2020.43,
  author =	{Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg},
  title =	{{Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.43},
  URN =		{urn:nbn:de:0030-drops-119046},
  doi =		{10.4230/LIPIcs.STACS.2020.43},
  annote =	{Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-F\"{u}rer-Immerman Graphs, coherent Configurations}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2017.2
Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

Authors: Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Abstract
Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X_1=(V,E_1) and X_2=(V,E_2) and a subset R\subseteq V\times V of forbidden pairs whether there is an isomorphism \pi from X_1 to X_2 such that i^\pi\ne j for all (i,j)\in R. We prove that this problem and several of its generalizations are in fact in \FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of R as parameters. The problem remains in FPT even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the number of moved vertices is unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism π between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that \pi can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT. In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [ESA 2011].
Cite as

Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán. Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arvind_et_al:LIPIcs.IPEC.2017.2,
  author =	{Arvind, Vikraman and K\"{o}bler, Johannes and Kuhnert, Sebastian and Tor\'{a}n, Jacobo},
  title =	{{Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{2:1--2:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.2},
  URN =		{urn:nbn:de:0030-drops-85690},
  doi =		{10.4230/LIPIcs.IPEC.2017.2},
  annote =	{Keywords: parameterized algorithms, hypergraph isomorphism, mislabeled graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.7
Parameterized Complexity of Small Weight Automorphisms

Authors: Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph. As a building block for our algorithms, we generalize Schweitzer's FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter. Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case.
Cite as

Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán. Parameterized Complexity of Small Weight Automorphisms. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2017.7,
  author =	{Arvind, Vikraman and K\"{o}bler, Johannes and Kuhnert, Sebastian and Tor\'{a}n, Jacobo},
  title =	{{Parameterized Complexity of Small Weight Automorphisms}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{7:1--7:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.7},
  URN =		{urn:nbn:de:0030-drops-70278},
  doi =		{10.4230/LIPIcs.STACS.2017.7},
  annote =	{Keywords: Parameterized algorithms, hypergraph isomorphism.}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.13
The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Authors: Vikraman Arvind, Frank Fuhlbrück, Johannes Köbler, Sebastian Kuhnert, and Gaurav Rattan

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
In this paper we study the complexity of the following problems: 1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard.
Cite as

Vikraman Arvind, Frank Fuhlbrück, Johannes Köbler, Sebastian Kuhnert, and Gaurav Rattan. The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2016.13,
  author =	{Arvind, Vikraman and Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Kuhnert, Sebastian and Rattan, Gaurav},
  title =	{{The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.13},
  URN =		{urn:nbn:de:0030-drops-64294},
  doi =		{10.4230/LIPIcs.MFCS.2016.13},
  annote =	{Keywords: parameterized complexity, graph automorphism, fixing number, base size, individualization}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2012.387
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace

Authors: Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky

Published in: LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Abstract
We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs; this important class of hypergraphs corresponds to matrices with the circular ones property. Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.
Cite as

Johannes Köbler, Sebastian Kuhnert, and Oleg Verbitsky. Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 387-399, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kobler_et_al:LIPIcs.FSTTCS.2012.387,
  author =	{K\"{o}bler, Johannes and Kuhnert, Sebastian and Verbitsky, Oleg},
  title =	{{Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)},
  pages =	{387--399},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-47-7},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{18},
  editor =	{D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.387},
  URN =		{urn:nbn:de:0030-drops-38757},
  doi =		{10.4230/LIPIcs.FSTTCS.2012.387},
  annote =	{Keywords: Proper circular-arc graphs, graph isomorphism, canonization, circular ones property, logspace complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2010.327
Colored Hypergraph Isomorphism is Fixed Parameter Tractable

Authors: V. Arvind, Bireswar Das, Johannes Köbler, and Seinosuke Toda

Published in: LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

Abstract
We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism} which has running time $2^{O(b)}N^{O(1)}$, where the parameter $b$ is the maximum size of the color classes of the given hypergraphs and $N$ is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm.
Cite as

V. Arvind, Bireswar Das, Johannes Köbler, and Seinosuke Toda. Colored Hypergraph Isomorphism is Fixed Parameter Tractable. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 327-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2010.327,
  author =	{Arvind, V. and Das, Bireswar and K\"{o}bler, Johannes and Toda, Seinosuke},
  title =	{{Colored Hypergraph Isomorphism is Fixed Parameter Tractable}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)},
  pages =	{327--337},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-23-1},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{8},
  editor =	{Lodaya, Kamal and Mahajan, Meena},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.327},
  URN =		{urn:nbn:de:0030-drops-28751},
  doi =		{10.4230/LIPIcs.FSTTCS.2010.327},
  annote =	{Keywords: Fixed parameter tractability, fpt algorithms, graph isomorphism, computational complexity}
}
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