Modular decomposition in kernelization › ~paul › Talks › talk-10-worker.pdf · 7 4 2 3...

Modular decomposition inkernelization

Christophe Paul

CNRS - LIRMM, Montpellier France

November 11, 2010 - worker 2010

Parameterized graph modification problems

I Can we update at most k vertices / edges of an input graphso get a property Π satisfied ?

Some examples

I FVS (or eq. treewidth 1 vertex deletion)

I odd cycle transversal

I minimum fill-in...

Parameterized graph modification problems

I Can we update at most k vertices / edges of an input graphso get a property Π satisfied ?

Some examples

I FVS (or eq. treewidth 1 vertex deletion)

I odd cycle transversal

I minimum fill-in...

Question:

Can modular decomposition help if we are interested inΠ-modification problems where Π is related to some width-measure

(clique-width, rank-width...) ?

Cluster editing as an appetizer

Modular decomposition in a nutshell

Cograph edge deletion kernel

Cluster Editing problem as an appetizer

Given a graph G = (V , E ) and an integer k, does there existF ⊆ V × V such that

I |F | 6 k and G M F is the disjoint union of cliques.


Structure of the solution

Observation 1: in each cluster, the non-affected vertices form a setof adjacent twins (or critical clique)



Observation 2: A critical clique C of size |C | > k cannot be spreadover several clusters



Observation 2: A critical clique C of size |C | > k cannot be spreadover several clusters

Rule: shrink every critical of size > k + 1 down to k + 1

⇒ O(k2) vertex kernel

More generally

Lemma Let F be an hereditary graph class closed under adjacenttwin addition. Then there exists an optimal solution to theF-edition problem that preserves the critical cliques.

More generally

Lemma Let F be an hereditary graph class closed under adjacenttwin addition. Then there exists an optimal solution to theF-edition problem that preserves the critical cliques.

For cluster editing

→ edit all or none of the edges between two critical cliques

→ edit the edge set of the critical clique graph

⇒ 2k vertex kernel [Guo]

A subset of vertices M of a graph G = (V , E ) is a module if∀x ∈ V \M, either M ⊆ N(x) or M ∩ N(x) = ∅

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Examples of modules

I critical cliques

I connected components

I connected components of G

I any vertex subset of thecomplete graph (or the stable)

I A graph is prime if all its modules are trivial: e.g. the P4.

I A graph is degenerate if any subset of vertices is a module:cliques and stables.

A partition P = {M1, . . . Mr} of the vertex set of a graph G is amodular partition of G if any part Mi ∈ P is a module of G .

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The quotient graph G/P is the induced subgraph obtained bychoosing one vertex per part of P.


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The quotient graph G/P is the induced subgraph obtained bychoosing one vertex per part of P.

Observation: The set of critical cliques form a modular partitionand the corresponding quotient graph is the critical clique graph.


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Lemma: X ⊆ P is a module of G/P iff ∪M∈XM is a module of G .

Theorem [Gal’67,CHM81] Let G be a graph. Then either

1. G is not connected, or


3. G/M(G) is a prime graph, with M(G ) the modular partitioncontaining the maximal strong modules of G .





A module is strong if it does not overlap any other module

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A module is strong if it does not overlap any other module

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inclusion tree of strong modules = modular decomposition tree

MD-tree can be computed in O(n + m)

Some references

I Mohring, Radermacher. Substitution decomposition fordiscrete structures and connections with combinatorialoptimization. Annals of Discrete Mathematics, 1984

I Habib, Paul. A survey on algorithmic aspects of modulardecomposition. Computer Science Review, 2010

I and many others...


G is a cograph if it can be built from the single vertex graph by:

I [disjoint union] G = G1 ⊕ G2

I [series composition] G = G1 ⊗ G2 (add all possible edgesbetween G1 and G2)

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G is a cograph if it can be built from the single vertex graph by:I [disjoint union] G = G1 ⊕ G2


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Observation: The cographs are exactlyI the clique-width 2 graphs

I the P4-free graphsI the totally decomposable graphs by the modular

decomposition


G is a cograph if it can be built from the single vertex graph by:I [disjoint union] G = G1 ⊕ G2


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Observation: The cographs are exactlyI the clique-width 2 graphsI the P4-free graphsI the totally decomposable graphs by the modular

decomposition



I |F | 6 k and G M F is a cograph).

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I use properties of modules

I use the (canonical) modular decomposition tree as witnessstructure of the solution

Observation: If M is a module and abcd is a P4, then either{a, b, c , d} ⊆ M or |{a, b, c , d} ∩M| = 1


We can work independently on

I subgraphs induced by modules and quotient graphs




Lemma If M is a module of the input graph G , then M is amodule of G M Fopt

edit all or none of the edge between two modules




Lemma If M is a module of the input graph G , then M is amodule of G M Fopt

edit all or none of the edge between two modules

→ same holds on tournaments for FAST

Rule: If M is a module of size at least k + 1 in G , then

I replace M by a clique module of size k + 1

I add G [M] as a connected component of the resulting graph




Observe the size of the graph does not decrease nor the parameter





Sunflower rule: if there are at least k + 1 edges pairwiseintersecting only on e, remove e and decrease the parameter.

Theorem [Guillemot, Perez, P.]:Cograph edge deletion has a cubic vertex kernel





Sunflower rule: if there are at least k + 1 edges pairwiseintersecting only on e, remove e and decrease the parameter.

Theorem [Guillemot, Perez, P.]:Cograph edge deletion has a cubic vertex kernel

Problem: improve the bound to quadratic

What to remember ?

I many graph properties can be defined / characterized byproperties of the modular decomposition tree

I the modular decomposition tree structure may provide aninteresting witness structure for the solution

I modules help us to separate / cut the problem we deal with

What to remember ?

I many graph properties can be defined / characterized byproperties of the modular decomposition tree

I the modular decomposition tree structure may provide aninteresting witness structure for the solution

I modules help us to separate / cut the problem we deal with

Modules already used in kernelization for:

I cluster editing, bicluster editing, min flip consensus tree,FAST, cograph edge modification. . .

What about generalizations ?

If M and M ′ are two overlapping modules then

I M \M ′ is a module

I M ∩M ′ is a module

I M∆M ′ is a module






The set of modules of a graph defines a partitive family






The set of modules of a graph defines a partitive family

Modular decomposition generalize to

I digraphs, permutations, k-structure...

I (weakly) partitive families, (weakly) bipartitive families (splitdecomposition)

I more generally tree-like set families

Example of problem related to these generalizations


I |F | 6 k and G M F is a distance hereditary graph




Theorem Let G be a graph. The following are equivalentI G is a distance hereditary graphI G is totally decomposable by the split decomposition

I G has rankwidth 1(BUT no finite set of forbidden subgraphs)

A split

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the split tree of a DH graph




Theorem Let G be a graph. The following are equivalentI G is a distance hereditary graphI G is totally decomposable by the split decompositionI G has rankwidth 1

(BUT no finite set of forbidden subgraphs)

A split

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the split tree of a DH graph

Modular decomposition in kernelization › ~paul › Talks › talk-10-worker.pdf · 7 4 2 3...

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