Confluent Data Reduction for Edge Clique Cover: A Bridge Between - - PowerPoint PPT Presentation

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Confluent Data Reduction for Edge Clique Cover: A Bridge Between - - PowerPoint PPT Presentation

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Introduction Clique Cover Graph transformation theory Partial Clique Cover Confluent Data Reduction for Edge Clique Cover: A Bridge Between Graph Transformation and Kernelization Hartmut Ehrig Claudia Ermel Falk H uffner Rolf

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Confluent Data Reduction for Edge Clique Cover: A Bridge Between Graph Transformation and Kernelization

Hartmut Ehrig Claudia Ermel Falk H¨ uffner Rolf Niedermeier Olga Runge

Technische Universit¨ at Berlin

2 September 2011

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Interaction of data reduction rules

Kernelizations typically use a set of data reduction rules Up to now, little research on interaction of reduction rules

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 2/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Interaction of data reduction rules

Kernelizations typically use a set of data reduction rules Up to now, little research on interaction of reduction rules

Definition

A set of data reduction rules is called confluent if any order of application yields the same instance.

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Why is confluence interesting?

If a kernel is confluent, it is “robust”; in an implementation, we can optimize for speed of application.

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Why is confluence interesting?

If a kernel is confluent, it is “robust”; in an implementation, we can optimize for speed of application. If a kernel is not confluent, it has “slack”: some orders might lead to worse results; investigating this might lead to improved rules.

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Why is confluence interesting?

If a kernel is confluent, it is “robust”; in an implementation, we can optimize for speed of application. If a kernel is not confluent, it has “slack”: some orders might lead to worse results; investigating this might lead to improved rules. Further, insights on the interaction between rules can lead to faster kernelizations.

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover

Clique Cover

Input: An undirected graph G = (V , E ) and an integer k 0. Question: Is there a set of at most k cliques in G such that each edge in E has both its endpoints in at least one of the selected cliques?

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover

Clique Cover

Input: An undirected graph G = (V , E ) and an integer k 0. Question: Is there a set of at most k cliques in G such that each edge in E has both its endpoints in at least one of the selected cliques?

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 4/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover

Clique Cover

Input: An undirected graph G = (V , E ) and an integer k 0. Question: Is there a set of at most k cliques in G such that each edge in E has both its endpoints in at least one of the selected cliques?

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 4/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover

Clique Cover

Input: An undirected graph G = (V , E ) and an integer k 0. Question: Is there a set of at most k cliques in G such that each edge in E has both its endpoints in at least one of the selected cliques?

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 4/20

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SLIDE 11

Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover

Clique Cover

Input: An undirected graph G = (V , E ) and an integer k 0. Question: Is there a set of at most k cliques in G such that each edge in E has both its endpoints in at least one of the selected cliques?

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 4/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Clique Cover

Rule 1

Delete isolated vertices.

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Clique Cover

Rule 1

Delete isolated vertices.

Rule 2

Delete isolated edges.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 5/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Clique Cover

Rule 1

Delete isolated vertices.

Rule 2

Delete isolated edges.

Rule 3

Delete one of two twins.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 5/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Clique Cover

Rule 1

Delete isolated vertices.

Rule 2

Delete isolated edges.

Rule 3

Delete one of two twins.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 5/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Kernelization for Clique Cover

Theorem ([Gy´ arf´ as 1990, Gramm et al. 2008])

Rules 1 to 3 yield a kernel with at most 2k vertices.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 6/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Confluence of Clique Cover kernel

Theorem

Rules 1 to 3 are confluent.

Proof

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 7/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Confluence of Clique Cover kernel

Theorem

Rules 1 to 3 are confluent.

Proof

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 7/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Confluence of Clique Cover kernel

Theorem

Rules 1 to 3 are confluent.

Proof

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 7/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Confluence of Clique Cover kernel

Corollary

A 2k -vertex kernel for CLIQUE COVER can be found in linear time.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 8/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Graph transformation theory

Started in the early 1970s Generalizes Chomsky grammars (on strings) and term rewriting systems (on trees) to graphs Used to model operational sematics of changing networks

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Reduction rules in graph transformation theory

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Reduction rules in graph transformation theory

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Reduction rules in graph transformation theory

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Reduction rules in graph transformation theory

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover reduction as graph transformation

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover reduction as graph transformation

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Clique Cover reduction as graph transformation

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Local confluence

Newman’s lemma [Newman 1942]

To show confluence of a system of data reduction rules, it is sufficient to show local confluence. G G1 G2 G3 Confluence ∗ ∗ ∗ ∗ G G1 G2 G3 Local confluence ∗ ∗

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Critical pair analysis

Theorem ([Plump 2005])

To show confluence of a system of data reduction rules on directed graphs, it is sufficient to consider critical pairs, that is, rule applications that conflict and have minimal context. G G1 G2 G3 Confluence of critical pair (G → G1, G → G2) ∗ ∗

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Critical pair analysis with AGG

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Partial Clique Cover

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Partial Clique Cover

Rule 4

Delete vertices incident only on covered edges.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 16/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Partial Clique Cover

Rule 4

Delete vertices incident only on covered edges.

Rule 5

Delete isolated edges.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 16/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Partial Clique Cover

Rule 4

Delete vertices incident only on covered edges.

Rule 5

Delete isolated edges.

Rule 6

Delete one of two twins when connections are labelled identically.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 16/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Data reduction for Partial Clique Cover

Rule 4

Delete vertices incident only on covered edges.

Rule 5

Delete isolated edges.

Rule 6

Delete one of two twins when connections are labelled identically.

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 16/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Kernel for Partial Clique Cover?

b1 b3 c1 c7

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Kernel for Partial Clique Cover?

b1 b3 c1 c7

Theorem

Rules 4 to 6 yield a kernel with at most 2k +c vertices, where c is the number of covered edges.

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Confluence of Partial Clique Cover rules

Theorem

Rules 4 to 6 are confluent.

Proof

u w = x1 x2 xm y1 yj . . .

1 1 1

R4(w) R6(u,v) R6(u,v) R4(w)

G2 G3 G1 G

... v

1

m ≥ 2 m ≥ 2

u x2 xm y1 yj . . . ... v u w = x1 x2 xm y1 yj . . .

1 1 1

... v

1 1 1 1

u x2 xm y1 yj . . . ... v

1 1 1

  • H. Ehrig et al. (TU Berlin)

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Future work and open questions

Kernelizations

Analyze more kernelizations for confluence Does it make non-existence proofs easier when only asking for confluent problem kernels? Does confluence help subsequent solution strategies that build on top of the kernel?

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 19/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Future work and open questions

Kernelizations

Analyze more kernelizations for confluence Does it make non-existence proofs easier when only asking for confluent problem kernels? Does confluence help subsequent solution strategies that build on top of the kernel?

Graph transformation theory

Extend critical pair theory to undirected graphs Extend critical pair theory to rule schemes Extend software tools with this

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 19/20

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Introduction Clique Cover Graph transformation theory Partial Clique Cover

Future work and open questions

Clique Cover

Is PARTIAL CLIQUE COVER in FPT wrt. k ? If so, does it have a singly-exponential kernel wrt. k ?

  • H. Ehrig et al. (TU Berlin)

Confluent Data Reduction 20/20