Isomorphism of (mis)Labeled Graphs

Schweitzer, Pascal

doi:10.1007/978-3-642-23719-5_32

Cited by 6 publications

(8 citation statements)

References 20 publications

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“…3. We observe that Schweitzer's algorithm in [24] can be used to compute, in k O(k) n O (1) time, the set M of all minimal support automorphisms σ ∈ Aut(X) such that |supp(σ)| ≤ k.…”

Section: Theorem 33 There Is a Kmentioning

confidence: 99%

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Arvind¹,

Fuhlbrück²,

Köbler³

et al. 2016

Preprint

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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 ⊂ 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 ≤ Sn given by generators, i.e., a minimum cardinality subset S ⊂ [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 (even in logspace), while starting from color class size 4 they become W[P]-hard.

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“…3. We observe that Schweitzer's algorithm in [24] can be used to compute, in k O(k) n O (1) time, the set M of all minimal support automorphisms σ ∈ Aut(X) such that |supp(σ)| ≤ k.…”

Section: Theorem 33 There Is a Kmentioning

confidence: 99%

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Arvind¹,

Fuhlbrück²,

Köbler³

et al. 2016

Preprint

View full text Add to dashboard Cite

show abstract

“…There are various results that show that isomorphism is fixed-parameter tractable with respect to some parameter. Such results exist for the parameters color multiplicity [18] (also known for hypergraphs [1]), eigenvalue multiplicity [13], rooted distance width [34], feedback vertex set number [25], bounded permutation distance [29], tree-depth [7] and connected path distance width [27]. For chordal graphs, tractability results are known for the parameters clique number [22], [26] and the size of simplicial components [32].…”

Section: Introductionmentioning

confidence: 99%

Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

Otachi

Schweitzer

2014

Algorithm Theory – SWAT 2014

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We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixedparameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions.As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.

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“…A different kind of constrained isomorphism problem was introduced by Schweitzer [15]. The weight (or support size) of a permutation π ∈ Sym(V ) is |{i ∈ V | i π = i}|.…”

Section: Introductionmentioning

confidence: 99%

“…We show that the problem is in FPT with t as parameter. Again, the "at most t" version of this problem was already shown to be in FPT by Schweitzer [15] as part of his algorithmic strategy to solve the weight at most k problem. Our results in Sections 4 and 6 also hold for hypergraphs when the maximum hyperedge size is taken as additional parameter.…”

Section: Introductionmentioning

confidence: 99%

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Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

Arvind,

Köbler,

Kuhnert

et al. 2017

Preprint

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Lubiw showed that several variants of Graph Isomorphism are NPcomplete, where the solutions are required to satisfy certain additional constraints [14]. 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 ⊆ V × V of forbidden pairs whether there is an isomorphism π from X 1 to X 2 such that i π = j for all (i, j) ∈ R. We prove that this problem and several of its generalizations are in fact in FPT:

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Isomorphism of (mis)Labeled Graphs

Cited by 6 publications

References 20 publications

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

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